www.BillHowell.ca's Background math for a review of Bill Lucas's book the "Universal Force, Volume I" /*/*$ echo "version= $date_ymdhm" >>"$p_augmented" version= 191025 17h43m This file is : /*/*$ echo "$p_augmented" >>"$p_augmented" /media/bill/ramdisk/191025 17h43m math Howell.txt /********************** >>> SUMMARY I've no doubt that results here are available in basic physics textbooks, but : it was important for me to work the material in detail in order to better understand the origins of key expressions this document provides a broad, detailed] basis to help users become familiar with "Howell's flatliner notation" equations parametric in t have been developed, both as a check on errors in my derivations, and to provide a the content is easily [edititable, auditable, extendible] once "Howell's flatliner notation" is understood. (see below) I've made several checks on taking a "cheating" apporoach to derivatives (for example, taking derivatives of scalar functions rather than their vector origins). Usually the cheating approach works fine. The level of detail provided allows the reader a clear view of my own errors and shortcomings. /*/*$ cat >>"$p_augmented" "$d_Lucas""context/summary - general.txt" As a warning, the material herein is EXTREMELY [repetitive, tedious, BORING], but it is a necessary evil to make sure. This document should be considered as a "first draft", as there are numerous errors and omissions, and to some degree it is an incomplete coverrage of "background math" for Lucas's Chapter 4. However, it is important to me as a detailed, step-by-step documentation, that is easily [auditable, editable, extendable], abeit in a specialised non-standard (yet more precise than conventional) system of [variable notations, symbols, notations, format]. /*_endCmd /*/*$ cat >>"$p_augmented" "$d_Lucas""context/text editor - how to set up.txt" +-----+ To view this file : - use a good text editor. I recommend kwrite (not kate) - do NOT use a word processor! That will likely corrupt the files, losing functionality. - set font as constant width (eg monospace), size = 10pt? - tabs are retained as tabs, not spaces - set tab width = 3 spaces - turn off word wrap - set auto-indents to the last tabbed position of a line - when necessary, use full screen mode to more easily see very long equations. +-----+ /*_endCmd *********************** TABLE OF CONTENTS /*_Insert_Table_of_Contents 153: SUMMARY 236: Lucas's Dedication 264: Introduction 346: I. Basics 350: [Observer, particle, ether] reference frames 379: Galilean transformation of the (observer, particle) reference frames, RFp <=> RFo 410: Generalized ether reference frames 446: Euclidean versus Riemannian geometries 546: Formulations of electrodynamics 551: Maxwell's equations 563: Covariant version 567: ?Heaviside? 4-vector formulation 571: ?Hamilton's? quaternion formulation 577: Lucas's equivalent 581: Ed Dowdye Jr's "Extinction shift principle" 585: Questions 589: Random, scattered questions 598: Initial linearity assumptions, but non-linear models 604: Superluminal speeds 616: Howell's use of the Kahan formulation for a "Scalar derivative of the norm of a vector function" 633: Time delays / Field lag - Temporal equivalence within a frame of reference for "short" distances? 643: How do toroidal [electrons, protons] behave with [spin,rotations, accelerations]? 653: Do toroidal [electrons, protons] [spin, oscillate] via [precession, obliquity]? 661: PROBLEM - When is an induce field "real"? 681: Is my application of Lenz's Law legitimate? 686: Major discrepancies between my own derivations and those of Lucas 690: BTodv(POIo,t)]" 779: BTpdv(POIo(t),t)] without use of Lenz's Induction Law (need to RE-CHECK!!!) 866: Lucas's vt*[cos(Aθpc(POIo(t),t)) - 1] term 875: B X v field is not electrostatic in nature 882: II. Derivations for a POIp = POIo(t) fixed in the particle reference frame (RFp) 887: Basic measures 890: Figure "Basic measures for for the particle reference frame RFp, using POIp=POIo(tx)" 911: Rpcv(POIp), Aθpc(POIp), Aφpc(POIp) are constants 925: Rpcv(POIo(t),t) 979: Rpcs(POIo(t),t) 1069: sin(Aθpc(POIo(t),t)) 1109: cos(Aθpc(POIo(t),t)) 1139: R_O0_pcs(POIo(t),t) 1169: RθPI2pcs(POIo(t),t) 1203: K0, K1, K2 for use in differentiations 1223: K0(t=0), K1(t=0), K2(t=0) in terms of E0ods(POIo,t) for relativistic factor, when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! 1254: EIods(POIo,t=0,ith stage) 1270: K_1st, K_2nd for use in differentiations 1338: K_1st(t=0), K_2nd(t=0), K_3rd(t=0) in terms of E0ods(POIo,t) for relativistic factor, when [t=0, RFp=RFo @t=0], use only AFTER differentiations!!! 1364: E0pdv(POIp) 1380: E0odv(POIo,t) 1392: B0pdv(POIp) = B0odv(POIo) = 0 1404: BIpdv(POIp) = 0 ≠ BIodv(POIp(t),t) = BIodv(POIo,t) 1420: BTpdv(POIp) = 0 1436: EIpdv(POIp) = 0 1444: ETpdv(POIp) = E0pdv(POIp) 1456: Derivatives 1460: Figure "Calculus for RFp, using POIp=POIo(t)" 1465: 1473: 1480: Aθpc(POIp)] = 0 1490: Rpcv(POIo(t) ] 1514: Rpcs(POIo(t),t) ]" 1607: Rpcs(POIo(t),t)] 1780: Rpcs(POIo(t),t)^(-α)] 1800: Aθpc(POIo(t),t)]" 1806: Aθpc(POIo(t),t)] 1911: Rpch(POIo(t),t)]" 1917: Rpch(POIo(t),t) ] 1986: sin(Aθpc(POIo(t),t))] 2098: cos(Aθpc(POIo(t),t))] 2235: Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))] = 0 2363: Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))] 2497: Rpcs(POIo(t),t)^(-b)*sin(Aθpc(POIo(t),t))^a] 2777: Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] 2848: Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] = 0 2923: K0,K2] for use in differentiations 2952: K1] for use in differentiations 2984: K0(t=0),K2(t=0)] in terms of E0ods(POIo,t) for relativistic factor, when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! 3013: K_1st,K_2nd,K_3rd] for use in differentiations 3069: K_1st(t=0),K_2nd(t=0),K_3rd(t=0)] in terms of E0ods(POIo,t) for relativistic factor, when [t=0, RFp=RFo @t=0], use only AFTER differentiations!!! 3101: E0pds(POIp)] 3111: E0ods(POIo,t)] 3137: Summary of ith stage EIods(POIo,t,ith stage)) derivations 3263: Summary of ith stage ETods(POIo,t,ith stage)) 3290: K_1st + f_sphereCapSurf(EIods(POIo,t=0,(i-1) stage))) ] 3302: E0ods(POIo,t)*sin(Aθpc(POIo(t),t))^a] 3359: E0ods(POIo,t=0)*Rpcs(POIo(t),t)^(-β)*sin(Aθoc(POIo))^a] 3502: E0odv(POIo)] 3512: BTodv(POIo,t)] 3528: Integrals 3531: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^z] 3577: Rpcs(POIo(t),t)^(-β)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a)] 3806: III. Derivations for a POIo = POIp(t) fixed in the observer reference frame (RFo) 3811: Basic measures 3814: Figure "Basic measures for the observer reference frame RFo, using POIo=POIp(tx)" 3833: Rocv(POIo), Aθoc(POIo), Aφoc(POIo) are constants in RFo 3851: Rocv(POIp(t),t) 3866: Rocs(POIp(t),t) 3896: RO0ocs(POIp(t),t) 3913: ROPI2ocs(POIp(t),t) = constant 3934: sin(Aθoc(POIp(t),t)) 3955: cos(Aθoc(POIp(t),t)) 4009: R_O0_ocs(POIo) 4018: E0odv(POIo,t) = E0pdv(POIo(t),t) ≠ E0pdv(POIp) = constant, except when t = tx 4083: E0ods(POIo,t) = E0pds(POIo(t),t) ≠ (for t ≠ tx) E0pds(POIp) = constant 4135: Figure "BTodv(POIo,t)" 4142: BIodv(POIo,t) = BIodv(POIp(t),t) ≠ BIpdv(POIp) = 0 4151: BTodv(POIo,t) = BTodv(POIp(t),t) ≠ BTpdv(POIp) = BTpdv(POIo(t),t) = 0 ???? 4164: Prediction of direction of field (B), given that the current I flows in the direction of the thum 4397: EIodv(POIo,t) = EIodv(POIp(t),t) ≠ EIpdv(POIp) = 0 4404: ETodv(POIo,t) = ETodv(POIp(t),t) = E0odv(POIp(t),t) + EIodv(POIp(t),t) ≠ ETpdv(POIp) = 0 4425: ETodv(POIo,t) = ETpdv(POIo(t),t) 4493: Derivatives 4496: Figure "Calculus for RFo, using POIo=POIp(t)" 4515: Aφoc(POIo)] = 0 4532: Rocv(POIp(t),t) ] 4951: 5162: Figure "Electrostatic field basics & calculus for a POIo" 5175: DEFINITIONS 5698: E0pds(POIo(t),t)] - cheating E0ods(POIo,t) scalar approach 5814: BTodv(POIo,t)]" 6229: Lenz's Induction Law - Basics and Calculus 6238: Lenz's Induction Law and it's context 6327: ET_LENZodv(POIo(t),t) = ET_LENZpdv(POIo(t),t), using Lenz's Induction Law 6393: EI_LENZods(POIo(t),t) = EI_LENZpds(POIo(t),t), using Lenz's Induction Law 6461: ET_LENZods(POIo(t),t) = ET_LENZpds(POIo(t),t), using Lenz's Induction Law 6516: BT_LENZodv(POIo(t),t) = BT_LENZpdv(POIo(t),t), using Lenz's Induction Law 6580: EI_LENZpdv(POIo(t),t)] - using Lenz's Law, proper vector approach 6810: ET_LENZpdv(POIo(t),t)], using Lenz's Induction Law 7022: BT_LENZpdv(POIo(t),t)] using Lenz's Induction Law, cheating scalar substitutions 7385: Basics and Calculus using Lucas's results from Thomas Barnes iterations 7389: Thomas Barnes iterations, it's context and relation to Lenz's Induction Law 7415: Howell's correction to the f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) iteration factor 7430: ET_BARNodv(POIo(t),t) = ET_BARNpdv(POIo(t),t), using Lucas's results from Thomas Barnes iterations (needs verification!) 7618: BT_BARNodv(POIo(t),t) = BT_BARNpdv(POIo(t),t), using Lucas's results based on Thomas Barnes iterations (needs verification!) 8442: IV. Expressions for the t=0 reference frame (RFt) 8454: Procedures for consistent reference frame switching 8469: Calculus of RFt 8509: APPENDICES 8512: Future extensions of the Universal Force 8527: Gaussian versus SI units ; 8541: Symbol checking and translation - short description 8627: HFLN = Howells FlatLiner Notation !!!!!!!!!!!!!! 31May2016 8647: Document build short description 8732: REFERENCES EQUATIONS : /*_Insert_equations For instructions on how to update the Table of Contents and Equations, see the section "Document build short description" at the end of this document. There is currently a problem of both lists above "shifting" the line number counts. /********************************************** waiver, copyright /*/*$ cat >>"$p_augmented" "$d_Lucas""context/waiver, copyright.txt" +-----+ Waiver/ Disclaimer The contents of this document do NOT reflect the policies, priorities, directions, or opinions of any of the author's past current, or future employers, work colleagues family, friends, or acquaintances, nor even of the author himself. The contents (including but not restricted to concepts, results, recommendations) have NOT been approved nor sanctioned at any level by any person or organization. The reader is warned that there is no warranty or guarantee as to the accuracy of the information herein, nor can the [analysis, conclusions, and recommendations] be assumed to be correct. The application of any concepts or results herein could quite possibly result in losses and/or damages to the readers, their associates, organizations, or countries, or the entire human species. The author accepts no responsibility for damages or loss arising from the application of any of the concepts herein, neither for the reader nor third parties. +-----+ Copyright © 2013 Charles W. Lucas, Jr. of Mechanicsville, Maryland, USA www.CommonSenseScience.org The book being reviewed, and formulae therefrom, are the property of Bill Lucas as indicated. Copyright © 2015 Bill Howell of Hussar, Alberta, Canada Exceptions: All papers cited are the property of the publisher or author as specified in the books and papers. All information from conversations with other individuals are potentially the property of that individual, or of third parties. Permission is granted to copy, distribute and/or modify ONLY the non-third-party content of this document under either: The GNU Free Documentation License (http://www.gnu.org/licenses/); with no Invariant Sections, Front-Cover Texts, or Back-Cover Texts. Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License. It is expected that users of the content will: Acknowledge William Neil Howell and/or the specific author of content or images on this page as indicated, as the source of the image. Provide a link or reference back to this specific page. Allow any modifications made to the content to also be reused under the terms of one or both of the licenses noted above. /*_endCmd /********************************************** >>> Lucas's Dedication (This is copied directly from his book.) /*/*$ cat >>"$p_augmented" "$d_Lucas""context/Lucas dedication.txt" Click to see Figure "Bill Lucas's 'Universal Force' book cover" http://www.BillHowell.ca/ Click to see /media/bill/SWAPPER/Lucas - Universal Force/Images/Lucas cover page.jpg This book is dedicated to all lovers of truth and especially the following : Euclid and the ancient Greeks that developed geometry and the axiomatic method to "Prove" or derive theories of natural philosophy in a systematic and logical way. Sir Isaac Newton who developed the empiracal scientific method to measure and mathematically define the minimal set of force equations to explain nature. James Clerk Maxwell who showed how to combine four of the six empirical laws of electRodynamics to develop his wave equations for electRodynamics which allowed the separate electric and magnetic force laws to be combined into a single electRodynamic force. He explained the wave nature of light which became the foundation of optics. He followed Michael Faraday and Andre-Marie Ampere in emphasizing the role of fields in extending ther electRodynamic force to great distances to replace Weber's action-at-a-distance electrodynamic force. Thomas L. Barnes, professor of Physics at the University of Texas at El Paso, who showed the way to eliminate Einstein's Special Relativity Theory from electRodynamics by taking into account the electrical feedback effects on finite-sized charged particles. Thomas Barnes, professor of physics at the University of Campinas - UNICAMP in Brazil, who showed the way to explain gravity as a fourth order electRodynamic effect between vibrating neutral electric dipoles using Weber's electRodynamic force. Alice Pittard Lucas my faithfiul and loving wife who encouraged and supported my research that resulted in this series of books. /*_endCmd /********************************************** >>> Introduction /*/*$ cat >>"$p_augmented" "$d_Lucas""context/introduction.txt" Since ?1967?, William Charles Lucas has been developing advanced concepts in electrodynamics with the intention of correcting for [simplifications, approximations, assumptions, incompleteness] that are inherent to the foundations of 20th century fundamental theoretical physics. Bill Lucas's book "The Universal Force, Volume I" revamps the theoretical basis of physics, starting with a re-formulation of electrodynamics that is built on concepts from [Newton, Maxwell, Barnes, Assiz,??], and a host of anomalous experimental data. Assuming that it is correct, the Universal Force avoids the [errors, inconsistencies, incompleteness] introduced by Maxwell's theorems, and immediately and simply eliminates any need for Lorentz-Poicare's theory of relativity or Einstein's later versions of Special and General relativity. It avoids the inherent [failures, mathematical contortions, complexity, contradictions] of relativity concepts, which can be viewed as a small subset of the Universal Force implications. Whole ecosystems of theories required to support and complete those theories also disappear, greatly reducing physics to a more complete "Natural Philosophy", and ensuring that concepts are "Theories should be made as simple as possible, but no more", to quote Einstein, and to also add that they should also work as more than band-aids and crutches. The volume also leads into structural physics, albeit leaving for a later volume a full description of how the Universal Force also replaces quantum mechanics. But is Lucas's work correct? To me, the old adage by ?name? "All theories are wrong, but some are useful" is a starting point, to which I add "While the most successful theories initially drive an explosion of [excitement, creativity, progress], they ultimately pass from science fashion, to science cult, to science religion, at which point mainstream scientists attack the dissidents [personally, professionally], and these mainstream concepts become the most important obstacle to our progress." So I prefer not to believe in belief nor absolute truth, and I do prefer to retain and compare "multiple conflicting hypothesis". In that light, Lucas easily passes the mark, to the extent that I have more confidence in his concepts than the mainstream. I am possibly helped in that perception by being too aware of the dark and ghastly details of the "past and present" of mainstream concepts and their bands of scientists and instituions. I delve into "correctness and proof" in more detail in my detailed review of Lucas's theoretical derivations. But at this stage my review ONLY addresses the correctness and consitency of his formula derivations, which to me is the starting point for having confidence in any deductive theory. The greater issue of agreement with data (and of the integretity and correctness of the data!) is not covered in my review, but for some points Lucas shows that key principles of mainstream science automatically fall out of the Universal Force, but that the latter is far richer and more powerful in going beyond mainstream concepts, both in specifics and for the unification of concepts. While at this interim stage my review is incomplete, and I do have remining questions about a coupel of important points that I am working on, it does give me great confidence in the thinking and work of Bill Lucas. Other priorities force me to put this aside for perhaps a year or two, even though there isn't much work left to complete the basic proofs of his derivations. If correct (or at least better), his work may : 1. extend and continue in the axiomatic manner established by Isaac Netwon, correct and extend James Clerk Maxwells work 2. simply erase the concepts of Max Planc and Albert Einstein. Of course, many of the well-researched and proven formulae will still remain, sometimes in improved form, as those are supported by data. 3. As Lucas shows in his book, one can often get to the correct formulae even if ones concept is incorrect. That gem is of great interest to me for my main priority outside of the area of Computational Intelligence (mostly neural networks) : to understand why and how the overwhelming mainstream scientific consensus and scientists fail so routinely, over [decades, hundreds, thousands] of years. My review of Lucas's core concepts is split among several documents : 1. "Howell - math of Lucas Universal Force.ndf" which goes step-by-step through Lucas's equations, providing "baby steps" to allow a careful verification of his results. 2. "Howell - Background math for Lucas Universal Force, Chapter 4.odt" This file contains my own derivations of background math related to Chapter 4 of Charles W. Lucas's book "The Universal Force, Volume 1". 3. "Howell - Background math, summary listing of Chapter 4 formulae.odt" provides a summary listing of comparisons between my pown "background math", and Lucas's versions of formulae as per his Chapter 4 equations. A short listing of serious discrepancies and issues is also provided. 4. Make a summary of serious discrepancies betweeen my reulsts and those of Lucas." 5. "Howell - Old math of Lucas Universal Force.ndf" Provides earlier verification attempts that went awry for several equations. In direct violation to the addage "It is better to remain silence and to have others think you a fool, ...", I make it very clear how foolish I can be. 6. "Howell - Verifications of Lucas Universal Force, summary listing .txt" This shows a quick summary listing of verification results for equations. All equations have much longer comments that can be seen in "Howell - math of Lucas Universal Force.ndf". 7. "Howell - Verifications of Lucas Universal Force, full listing .txt" This shows a more detailed summary listing of verification results for equations. All equations have much longer comments that can be seen in "Howell - math of Lucas Universal Force.ndf". 8. "Howell - Symbols for Bill Lucas, Universal Force.pdf" Beyond a listing of Lucas's "variable symbols and notations", which was a great reminder for me during my verification pRocess, this document also provides a description of my own non-standard format for [equations, array & vector notations, basic operations like integration & differentiation]. This will probably be essential for readers of "Howell - math of Lucas Universal Force.ndf" 9. "Howell - Review of Lucas, Universal Force.pdf" Peer-review style comments on the contents of the book and its concepts, including my perceptions of its strengths, weaknesses, and questions that I have. 10. "Howell - Meta-Level Lucas Universal Force speculative [context, comments, questions] - random, scattered blah, blah from yours truly. (unwritten as of 24Sep2015) 11. "Howell - the twin brothers [Science, Religion] and their disciples" There is an uncany resemblance between groups that often see themselves as polar opposites. Comments of a science fiend but non-believer of either. (unwritten as of 24Sep2015) This document provides my own verifications of Lucas's equations as described in an older version of his book "The Universal Force: Volume 1" (Lucas2013?). References for this verification are provided the section "REFERENCES" below. My work here serves primarily to force myself to look closely at his work, to better form my opinions on its veracity and potential. My own background is highly mixed, involving varying degrees of Engineering, Research management, marketing, business development etc. I am certainly NOT an expert in physics, and this is a learning experience for me even if some of it is a repaeat of long-forgotten [math, physics, engineering] courses. I have followed Lucas's basic formulations, even though some of his expressions for key relationships do not seem to agree with standard literature expressions, such as : (4-1) Generalised Amperes Law (4-5) Lenzs Induction Law (4-6) Lorentzs Force Law 17May2016 HOWEVER, I still have to go through Appendix A for the Generalized Amperes Law. Comments on approach : As I worked through Lucas's equations, at times I obtained somewhat different results. However, in going to the next step, in general (not always) I used his version of the result. I have a high degree of confidence in his knowledge and work now, more than my own limited grasp of the area, even if I still feel there is a chance of occasional errors or differences of inter´retations that may ultimately arise with his theories. This approach also makes it possible to "skip to" different ewquations out of sequence, even though I have not done so with Chapter 4. My [style, approach] in this math review has developed.over.the years for doing my peer reviews of papers (mostly nerial network related) that present new mathematical theorems anbd proofs. My standard comment inmy reviews also applies to the current document : "... As a reviewer, I find that a step-by-step re-typing of a part of the paper as I have done below forces me to pay attention to details that I might otherwise skim over. Even though this is perhaps too time intensive to apply to the full paper, by doing so.over.part of the authors work, it gives me far grater confidence in the rest of the paper, which is read, but not analysed step-by-step. It also gives the authors a better idea of the weaknesses of the reviewer! ..." One thing different about the current review is that I have saved "older versions" of my attempts to verify Lucas's in a separate file "Howell - Old math of Lucas Universal Force.ndf" where I have deliberately left in my mistakes, wrong turns, dead ends etc, rather than clean it all up. No doubt this will be of great annoyance to the reader of that file, but its very important to me, as it highlights my own weaknesses, and serves as a reminder that things dont always flow as smoothly as I would like. Others who wish to criticise my work and weaknesses will also find it useful, and Im all for criticism. Further more, some of the dead ends have lessons to teach. Im a big fan of "Multiple Conflicting Hypothesis" and "Multiple Independent Pathways" to get to an answer, almost in a Category Theoretical sense (no - Im no good at Category Theory, but I find it very intriguing). This file is written so that it is suitable to be loaded directly by the QNial programming language. Most of the content is commented out (text without empty lines following the character # when it is the first character in a line). For now this facilitates providing up-to-date summaries of results for each equation, and checking of any numerical calaculations. Some degree of symbolic processing is also possible, but not necessary as of 07Sep2015. Most of my symbolic processing code is for formulas subject to array computations, where the formulae are processed, not numbers. However, Lucas's book does not get into tensor notations and processing - but this may become important at a later stage when going through standard General Relativity work. www.BillHowell.ca 10Sep2015 /*_endCmd /********************************************** >>> I. Basics /***************************************** >>>>>> [Observer, particle, ether] reference frames /*/*$ cat >>"$p_augmented" "$d_Lucas""context/reference frames.txt" ...need to rewrite!! ... ??? show Figure of "capped" sphere ??? For Chapter 4, Lucas chooses an observer Frame of Reference (RFo) that is of the same [scale, orientation] as the particle Frome of Reference (RFp), and that is initially coincident with it (t=0). Furthermore, the particle's velocity within RFo is constant This section deals , whereas all other points in space (or Points Of Interest (POI) including the observer, are fixed with resepct to RFo. This vastly simplifies the geometry and formulations. Note that within RFp, given symmetry and constant observer velocity, Et,E0,Ei, and Bt,B0,Bi are only functions of [rp,Op,t] within the PARTICLE reference frame (eg Et(rp(t),Op(t)). The time variance is useful for following changes in [B,E,F, etc] of a fixed "Point of interest" in the particle & observer reference frames. ??? show Figure of plane defined by the particle center and Lo ??? Refering to Figure ??? "capped" sphere ??? : Figure "Chapter 4 reference frames" This sub-section contains a "partially modified symbol" version of my initial Sep2015 analysis, which includes a mix of Cartesian, Spherical] coordinate expression, and which is the basis for later "General reference frames" work. It should all be updated to my current symbol definitions, but it's actually easier to redo the derivations rather than modify the old. For Chapter 4, changes in [E,B,...] occur in RFo, not RFp, so it is important to derive expressions for a Point Of Interest (POI) that is fixed in the observer reference frame (RFo). For this case, particle reference frame (RFp) calculations of [E,B,...] will vary with time for the POI, unlike the previous section, whereas observer coordinates to the POI are now constant. /********************* >>>>>>>>> Galilean transformation of the (observer, particle) reference frames, RFp <=> RFo For Chapter 4, Lucas chooses a Reference Frame for the observer (RFo) that is of the same [scale, orientation] as the Reference Frame for the particle (RFp), and that is initially coincident with it (t=0). Furthermore, the particle's velocity within RFo is constant For Chapter 4, symmetry around Roch, and a constant particle velocity along a coordinate system axis, makes Pp irrelevant to most POIp measures and calculations here. +-----+ 27Mar2018 WRONG!! As the POIp moves with the particle : [Rpcs,ROpcs,ROPI2pcs,Opc,Ppc,Poc,Ep,Bp] are NOT functions of time. [Rocs,ROocs,ROPI2ocs,Ooc, Eo,Bo] are functions of time. +-----+ For illustrations of the geometries and symbols, see the illustrations and definitions in "Howell - Symbols for Bill Lucas, Universal Force.odt". OBJECTIVES : Where convenient, express variables as functions of FIXED variables : (POIp) -> -Vodv(PART),t,Rpcv(POIp),Aθpc(POIp),Aφpc(POIp) (POIo) -> Vodv(PART),t,Rocv(POIo),Aθoc(POIo),Aφoc(POIo) To "translate" (transform) RFo to RFp coordinates, the Galilean transformationapplies. See the Figure "Galilean transformation" below for details. That the [observer, particle] reference frames are of the SAME [rotation, scale], and are exactly coincident at time t=0, makes [analysis, formulae] MUCH easier !!!! Figure "Galilean transformation" Click to see http://www.BillHowell.ca/ Click to see file:///media/bill/SWAPPER/Lucas - Universal Force/Images/Howell - Chapter 4 - Galilean transformation - cropped.png /********************* >>>>>>>>> Generalized ether reference frames While the [observer, particle] reference frames seem clear enough, for general reference there may be a need to define a "Generalized ether reference frame" that will likely be completely independent of the [observer, particle] frames. At this stage of my Chapter 4 work (20Dec2017), I have not yet elaborated on systems of reference frames for "generalized ether frameworks". As such, my current checks on derivations for Chapter 4 simply follow Lucas's own interpretations. Lucas does incorporate the Lorentz-Poincare relativistic correction factor (adopted by Einstein, but which interferometer data clearly contradict[?2012 Rob Johnson?]), even though he uses Thomas Barnes approach to deriving the factor via classical physics rather than relativity theory. It seems that essentially all physicists seem "conceptually stuck" with one simple ether concept from the start of the discussions about the apparently anomalous speed of light, very, very long ago, It seems that only rare physicists are [interested, willing, able] to consider the vastly different concepts, or to realize that General Relativity itself can be interpreted s incorporating an ether concept. As a quick, inaccurate list, of ether fields from simplest to more powerful (albeit not necessarily more correct) : 1. there is no ether, as the ether concept was a mistake of applying to strictly [solid, liquid, gas, plasma] wave physics to light 2. material 3. electromagnetic 4. gravitational 5. General relativity - While initially Einstein and colleagues made great ado about abolishing ether, it seems that they later realized that General Relativity DOES incorporate ether-like concepts, but they decided to avoid the use of the word "ether" and the great about-face they had done? [many references – list??]. 6. Zero point energy 7. PLUS a seemingly endless series of other suggestions for ether 8. Any combinations of the above 9. Note that for some concepts, some of the concepts above are a manifestation of other concepts, such as : Satellite atomic clocks from a conventional point of view – clock rates are [solar activity, altitude]-dependent, General relativity Zero-point energy Lucas's "Universal force" : gravity is simply the net attractive 4th order electromagnetic force between neutral vibrating dipoles (similar to Van der Waal's force in chemistry) material (mass-related) is a manifestation of electromagnetic energy, involving Mach's principle 10. Note that [Ether - particle – electromagnetic] interactions are not discussed here It is very important to specify assumed motions of an ether with respect to one or more of the [Earth, Sun, local region of Milky Way, Milky Way center, Center of Universe] : 1. non-rotational movement at a constant rectilinear velocity 2. rotating at a constant angular velocity 3. accelerating [translational (recti-linear), rotational, perhaps even sub-atomic spin] - I don't remember any examples of this beyond GR, but I haven't checked. 4. Point 3 for systems that are multi-scalar [periodic, quasi-periodic, chaotic] and/or [coupled, synchronous, entangled...] For disciples of the great science [fashion-cum-cult-cum-religion] of the Big Bang Theory and its expansion of the universe would seem to scream for combinations of perhaps all of the concepts above. /********************* >>>>>>>>> Euclidean versus Riemannian geometries https://en.wikipedia.org/wiki/Non-Euclidean_geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is replaced with an alternative one. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry. The essential difference between the metric geometries is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line ℓ and a point A, which is not on ℓ, there is exactly one line through A that does not intersect ℓ. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting ℓ, while in elliptic geometry, any line through A intersects ℓ. Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line: In Euclidean geometry the lines remain at a constant distance from each other (meaning that a line drawn perpendicular to one line at any point will intersect the other line and the length of the line segment joining the points of intersection remains constant) and are known as parallels. In hyperbolic geometry they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels. In elliptic geometry the lines "curve toward" each other and intersect. https://www.quora.com/How-do-I-explain-the-difference-between-Euclidean-geometry-and-Riemannian-geometry-to-a-curious-and-intelligent-15-year-old How do I explain the difference between Euclidean geometry and Riemannian geometry to a curious and intelligent 15 year old? Rahul Anand, Work smart, Live hard and Enjoy easy Written May 25, 2016 Briefly speaking Euclidean Geometry is the study of flat spaces. In case you have noticed all the axioms and the postulates are mainly dedicated to 2-dimensional. There are a few exceptions. Now, Riemannian Geometry is an example of the non-euclidean geometry( There are forms of geometry that contain a postulate (axiom) which is equivalent to the negation of the Euclidean parallel postulate. ) Simply saying Riemannian Geometry is the study of curved surfaces. Hope you understood it finally. Cheers http://mathforum.org/library/drmath/view/64459.html Euclidean and Riemann Geometry Date: 12/29/2003 at 01:33:36 From: Ming Subject: Riemann Geometry I'm a bit confused about the basic premises of Riemann Geometry. I want to write a paper comparing it to the basic premises of Euclidean Geometry and do a comparison on how they differ, and how that subsequently affects our understanding of geometry and the natural world. What I've done so far is discuss the first 4 axioms of Euclidean Geometry, then talked about why the fifth axiom is more a statement of fact than it is an axiom, and why people have tried to prove it but failed. Then I move on to Riemann's work. Here is where the problem arises--I understand that he changed it, saying that parallel lines always meet. But how did he come up with this, and why is it true? Date: 12/30/2003 at 07:46:56 From: Doctor Edwin Subject: Re: Riemann Geometry Hi, Ming. This is a fun topic. I like it because it illustrates a lot about how math relates to the world. Really, geometry isn't about the world. It's about geometry. It's a nice, closed system. But of course the entities in geometry are a lot like things in the real world, aren't they? And that's what makes it useful. Euclid built a closed system that was similar to the way things behave in the real world. But as you pointed out, there were lots of things that were true in the real world that couldn't be derived in geometry with just the four postulates. So Euclid took a thing that seemed true in the real world and added it to his system as a fifth postulate. Then he was able to have a pretty complete model of how shapes worked in the real world. Many people (including, I seem to recall reading, Euclid) were unhappy with the fifth postulate and tried to get rid of it. If you could derive it from the other four postulates, you could have it as a theorem, and be back to the original four. One way to prove the fifth postulate would be by negation. If I assume the opposite of the fifth postulate, and add that to the system, and I can find a way that it contradicts one of the other four postulates, then I have proven that the fifth postulate is true, using the other four. There are two ways to negate the fifth postulate. You can either assume that there are NO lines through a point not on line AB that are parallel to AB, or that there are an infinite number of them. Riemann's geometry assumes that there are no parallel lines--that all lines must intersect. However, when this was done, no contradiction was found. You could generate theorems using the negation of the fifth postulate along with the other four from now until the cows come home, and you'd have nothing but a perfectly self-consistent closed system that is about itself. You'd have an alternate geometry. Okay, here's the cool part. Just like Euclid's geometry models the way shapes work on a plane, Riemann's geometry models the way shapes work in a space that curves back on itself, like on the surface of a sphere. Now here's the really cool part. Einstein said that the universe actually fits Riemann's geometry--that the three-dimensional universe we perceive actually curves back on itself in four dimensions like the two-dimensional surface of a sphere does in three dimensions. Does that help answer your question? /*_endCmd /***************************************** >>>>>> Formulations of electrodynamics It is important for me to emphasize several key [equations , systems of models] that will be well known to experts in the area, for those like me who tend to forget. /********************* >>>>>>>>> Maxwell's equations (... I need to explain the differences between the classic 4-vactor Maxwell equations, and those of Lucas" ...) perhaps provide a table... 31Mar2016 Big questions at the base : Normally, Maxwell's equations relate RATE OF CHANGE of B to E : as with (4-2) Faraday'sd Law, but not like (4-1) Generalized Ampere's Law (actulally latter probably OK?) Lenz's Law - is KEY, need proof from other approaches as this is a HUGE simplification! Barnes iterations - how does this look fundamentally-geometrically? /********************* >>>>>>>>> Covariant version Jackson 1999 p??h?? Equation ?? /********************* >>>>>>>>> ?Heaviside? 4-vector formulation /********************* >>>>>>>>> ?Hamilton's? quaternion formulation Can one consider this to be the proper basis? /********************* >>>>>>>>> Lucas's equivalent /********************* >>>>>>>>> Ed Dowdye Jr's "Extinction shift principle" /***************************************** >>>>>> Questions ******************** >>>>>>>>> Random, scattered questions 1. Is the particle frame of reference adequate for a "point of interest" (observer or particle reference frame) when charges have finite size, or is a "Point of Interest" reference frame required to make calculations tractable? 2. With distributed charges, even within the particle frame of reference calculations become much more challenging. 3. Do Lucas's formulae properly integrate the contributions of distributed charge? He assumes toroidal loops, but the calculations seem to show only one of infinitely many possible arranagments of two toroidal loops, and don't show the effect of neutral dipols within U atoms, or molecular compounds of much greater size and complexity. 4. After a lifetime of being "programmed" to believe in a Dirac Nucleus and electron shell, is there a better way to transition to Lucas's point of view to better understand it? 5. I am guessing somewhat at several details of the separate [particle, observer] frames of reference. /********************* >>>>>>>>> Initial linearity assumptions, but non-linear models From Ampere, Faraday, Oersted, etc - basics "Laws" are linear. But the Universal Force (and Jeffimenko's causality, probably Randall Mills) is non-linear, meaning that the derivations are self-inconsistent? /********************* >>>>>>>>> Superluminal speeds IMPORTANT NOTE : There is an explicit belief in mainstream physics, AND IN LUCAS'S book, an implicit assumption, that speeds can never exceed the speed of light in a vacuum, c. However, that should NOT be a limitation of Lucas's Universal force!! There is therefore no guarantee that v/c is <1, and therefore that : Nyet : 0 <= [(1 - lambda(v)), (1 - beta^2) <= 1 This affects derivatives! /********************* >>>>>>>>> Howell's use of the Kahan formulation for a "Scalar derivative of the norm of a vector function" See below : "Scalar derivative of the norm of a vector function" : In working with vector derivatives, I have blindly made a key assumption regarding the Kahan formulation : d||z|| = u_T dotPRod dz / ||z|| where u_T is the linear functional dual to z wrt ||...|| But is this correct? Or, more specifically, under what conditions will I run into trouble with this assumption? Given the importance of vector derivatives, I really need to look into this much more closely! Likewise, the same question applies to my use of "Scalar integral of the norm of a vector function" : Therefore, the integral is : |x| = ∫[dx : x/|x| } for x≠0. /********************* >>>>>>>>> Time delays / Field lag - Temporal equivalence within a frame of reference for "short" distances? For a particle moving through space at constant velocity with no interactions (eg [E,B, other forces) along the way, we will assume that the [E,B] field structures originating from the particle do not change, with an important limitation. If one assumes that the fields extend instantaneously to the ends of the universe, then there is no need to consider "field lag". This seems to be Lucas's assumption. However, assuming a speed of propagation of electromagnetic fields to equal that of electromagnetic radiation then one must consider field lag. For the purposes of Chapter 4, we'll assume that rps_max is limited in the analysis to a scale within which lag effects are negligible, which also means that the particle motion has been in effect long enough BEFORE we start the analysis to establish the "field lag" to the extent of rs. /********************* >>>>>>>>> How do toroidal [electrons, protons] behave with [spin,rotations, accelerations]? Given the toroidal ring model favoured by Lucas and his sources [?Bergman, ???], what happens when a single [electron, proton, atom, molecule, clumped of matter] are subject to non-rectilinear accelerations? My first guess is that gyroscopic forces of some sort would be involved, and would tend to cancel one another in a clump of solid matter consisting of large numbers of molecules in random orientations. Neutral atomes might have the same property, with gyroscopic forces balancing out? Might this relate to [polarization, spin, color] in subatomic physics? /********************* >>>>>>>>> Do toroidal [electrons, protons] [spin, oscillate] via [precession, obliquity]? For ?names - Bergman, etc??? toroidal [electron, proton] structural theory favoured by Lucas, do gyroscopic-like restraints allow for free spin of the atom? Presumably at least oscillation occurs? Note that recently, Lucas has claimed that recent NIST data shows that neutrons are NOT a different sub-atomic particle, and are merely a tightly bound electron-proton pair. This is reminiscent of the beta particle (helium nucleus). /********************* >>>>>>>>> PROBLEM - When is an induce field "real"? see the 6 equations in Lucas p64 This is a question I have from equations [(4-13),(4-14),(4-15)] /* Generalized Ampere's Law -> "v" will be very different for different reference frames. - If I assume that the Bi field is the same in all reference frames, then v is a velocity relative to WHAT? - Presumably it must be relative to the E0 field? From this, I conclude that Equation 1 - implies that the B field DOES depend on the observer frame! So if : - you moved with a point charge, you would see NO B induced field? - a very high-speed passing observer would see arbitrarily large B fields? - many observers moving past the point at different relative velocities will see different B fields Weird, I didn't think of these things. Reminds me that a constant current in a wire has a B field, but no E field (which I hadn't thought of)! /********************* >>>>>>>>> Is my application of Lenz's Law legitimate? /***************************************** >>>>>> Major discrepancies between my own derivations and those of Lucas /********************* >>>>>>>>> Form of expression for "∂[∂(t): BTpdv(POIo(t),t)] = ∂[∂(t): BTodv(POIo,t)]" /*/*$ cat >>"$p_augmented" "$d_augment""d-dt BTpdv vs BTodv discrepency with Lucas.txt" /*+-----+ From Lucas's book : 10Jan2017 corrections p68h0.0 Equation (4-16), with my more up-to-date symbols, and removing redundant "c"'s (and using BT = B0 + BI, but B0 = 0) : /%(4-16) ∂[∂(t): BTpdv(POIo(t),t)] = ∂[∂(t): BTodv(POIo,t)] = Vons(PART)/c*Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^5*[Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) - Vons(PART)*t] - ∂[∂(t): EIpds(POIo(t),t)]/Rpcs(POIo(t),t) } = Vons(PART)/c*Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^5*Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) - 3*Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^5*Vons(PART)*t - ∂[∂(t): EIpds(POIo(t),t)]/Rpcs(POIo(t),t) } = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*Q(PART)*Vons(PART) *cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 - 3*Q(PART)*Vons(PART)^2*t /Rpcs(POIo(t),t)^4 - ∂[∂(t): EIpds(POIo(t),t)] } = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*Q(PART)*cos(Aθpc(POIo(t),t)) /Rpcs(POIo(t),t)^3 - 3*Q(PART)*Vons(PART) *t/Rpcs(POIo(t),t)^4 - ∂[∂(t): EIpds(POIo(t),t)]/Vons(PART) } = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*Q(PART) /Rpcs(POIo(t),t)^3 - 3*Q(PART) /Rpcs(POIo(t),t)^3 *Vons(PART) *t/cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) - ∂[∂(t): EIpds(POIo(t),t)]/Vons(PART)/cos(Aθpc(POIo(t),t)) } ∂[∂(t): BTpdv(POIo(t),t)] = ∂[∂(t): BTodv(POIo,t)] = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*Q(PART) /Rpcs(POIo(t),t)^3 *[1 - Vons(PART)*t/cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) ] - ∂[∂(t): EIpds(POIo(t),t)]/Vons(PART)/cos(Aθpc(POIo(t),t)) } Summarizing : (1) ∂[∂(t): BTpdv(POIo(t),t)] = ∂[∂(t): BTodv(POIo,t)] = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*Q(PART)/Rpcs(POIo(t),t)^3 *[1 - Vons(PART)*t/cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) ] - ∂[∂(t): EIpds(POIo(t),t)]/Vons(PART)/cos(Aθpc(POIo(t),t)) } /*+--+ 15&16May2016 But looking at : /% Vons(PART) *t/cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) From "Rθ0pcs(POIo(t),t)", RFo basis : (1)* Rθ0pcs(POIo(t),t) = Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t From "Rθ0pcs(POIo(t),t)", RFp basis (1)** Rθ0pcs(POIo(t),t) = Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) combining (1)* & (1)** : Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t = Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) So : Vons(PART)*t = Rocs(POIo)*cos(Aθoc(POIo)) - Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) or : Vons(PART)*t/cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) = Rocs(POIo)*cos(Aθoc(POIo))/Rpcs(POIo(t),t)/cos(Aθpc(POIo(t),t)) - 1 (2) = Rθ0ocs(POIo)/Rθ0pcs(POIo(t),t) - 1 Subbing (2) into (1) : (1) ∂[∂(t): BTpdv(POIo(t),t)] = ∂[∂(t): BTodv(POIo,t)] = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*Q(PART) /Rpcs(POIo(t),t)^3 *[1 - Vons(PART)*t/cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) ] - ∂[∂(t): EIpds(POIo(t),t)]/Vons(PART)/cos(Aθpc(POIo(t),t)) } = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*Q(PART) /Rpcs(POIo(t),t)^3 *[ 1 - Rθ0ocs(POIo)/Rθ0pcs(POIo(t),t) - 1 ] - ∂[∂(t): EIpds(POIo(t),t)]/Vons(PART)/cos(Aθpc(POIo(t),t)) } Summarizing : (3) ∂[∂(t): BTpdv(POIo(t),t)] = ∂[∂(t): BTodv(POIo,t)] = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*Q(PART) /Rpcs(POIo(t),t)^3 *Rθ0ocs(POIo)/Rθ0pcs(POIo(t),t) - ∂[∂(t): EIpds(POIo(t),t)]/Vons(PART)/cos(Aθpc(POIo(t),t)) } /********************* Compare this to : /********************* >>>>>>>>> ∂[∂(t): BTodv(POIo,t)] = ∂[∂(t): BTpdv(POIo(t),t)] without use of Lenz's Induction Law (need to RE-CHECK!!!) /*Howell's expression Section "∂[∂(t): BTodv(POIo,t)] = ∂[∂(t): BTpdv(POIo(t),t)] without use of Lenz's Induction Law (need to RE-CHECK!!!)" /%(6)* ∂[∂(t): BTpdv(POIo(t),t)] = ∂[∂(t): BTodv(POIo,t)] = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*Q(PART) /Rpcs(POIo(t),t)^3 11Jan2017 WRONG UNITS!!! - EIpds(POIo(t),t)/Rpcs(POIo(t),t) - ∂[∂(t): EIpds(POIo(t),t)]/Vons(PART)/cos(Aθpc(POIo(t),t)) } /*While the factor of 3 IS EXPLAINED, this is NOT the same as (3) in the sub-sub-section immediately above, as I have in (6)* an extra term for ∂[∂(t): E0] at right angles to Rodh(Vonv_X_Rpcv(POIo)), due to angular changes. However, it's hard to tell given the term EIpds(POIo(t),t)/Rpcs(POIo(t),t). By applying Lenz's Law : From "Lenz's Induction Law and it's context" based on Lucas p70h0.85 Equation (4-31) : /%(4-31) EIpds(POIo(t),t)*Rpch(POIo(t),t) = -λ(Vons(PART))*E0ods(POIo)*Rpch(POIo(t),t) and therefore : EIpds(POIo(t),t) = -λ(Vons(PART))*E0ods(POIo) From "E0odv(POIp(t),t) = E0pdv(POIp)" : (1)* E0pdv(POIp) = E0odv(POIp(t),t) = Q(PART)/Rpcs(POIp)^2*Rpch(POIp) Subbing (1)* into (4-31)* : (4-31) EIpds(POIo(t),t)*Rpch(POIo(t),t) = -λ(Vons(PART))*E0ods(POIo)*Rpch(POIo(t),t) (1) = -λ(Vons(PART))*Q(PART)/Rpcs(POIp)^2*Rpch(POIo(t),t) Subbing (1) into (6)* : ∂[∂(t): BTpdv(POIo(t),t)] = ∂[∂(t): BTodv(POIo,t)] = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*Q(PART) /Rpcs(POIo(t),t)^3 - EIpds(POIo(t),t)/Rpcs(POIo(t),t) - ∂[∂(t): EIpds(POIo(t),t)]/Vons(PART)/cos(Aθpc(POIo(t),t)) } = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*Q(PART) /Rpcs(POIo(t),t)^3 - (-λ(Vons(PART))*Q(PART)/Rpcs(POIp)^2)/Rpcs(POIo(t),t) - ∂[∂(t): EIpds(POIo(t),t)]/Vons(PART)/cos(Aθpc(POIo(t),t)) } /*Do I have a mistake with the sign here? /*Summarizing to yield Howell's expression Section "∂[∂(t): BTodv(POIo,t)] = ∂[∂(t): BTpdv(POIo(t),t)] subbing for Lenz's Law : /%(2) ∂[∂(t): BTpdv(POIo(t),t)] = ∂[∂(t): BTodv(POIo,t)] = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*Q(PART) /Rpcs(POIo(t),t)^3 *[1 + λ(Vons(PART))/3] - ∂[∂(t): EIpds(POIo(t),t)]/Vons(PART)/cos(Aθpc(POIo(t),t)) } /*Substituting for lambda from Lucas p73h0.7 in the text : /% lambda = β^2 = (Vons(PART)/c)^2 /*+-----+ REPEATING results for ease of comparison : From Lucas (4-16), sub-sub-section above : /%(3) ∂[∂(t): BTpdv(POIo(t),t)] = ∂[∂(t): BTodv(POIo,t)] = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*Q(PART) /Rpcs(POIo(t),t)^3 *Rθ0ocs(POIo)/Rθ0pcs(POIo(t),t) - ∂[∂(t): EIpds(POIo(t),t)]/Vons(PART)/cos(Aθpc(POIo(t),t)) } /*Howell's expression Section "∂[∂(t): BTodv(POIo,t)] = ∂[∂(t): BTpdv(POIo(t),t)] subbing for Lenz's Law : /%(2) ∂[∂(t): BTpdv(POIo(t),t)] = ∂[∂(t): BTodv(POIo,t)] = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*Q(PART) /Rpcs(POIo(t),t)^3 *[1 + λ(Vons(PART))/3] - ∂[∂(t): EIpds(POIo(t),t)]/Vons(PART)/cos(Aθpc(POIo(t),t)) } /*These are "somewhat similar", and the factor of 3 is explained. However, there are very important differences, as highlighted above. As for the cause of the discrepancy, that is not fully clear : As per above "... While the factor of 3 IS EXPLAINED, this is NOT the same as (3) in the sub-sub-section immediately above, as I have in (6)* an extra term for ∂[∂(t): E0] at right angles to Rodh(Vonv_X_Rpcv(POIo)), due to angular changes. ..." perhaps there is also a difference between point-particle and finite-sized derivations, as I have not yet done the latter? But this is not obvious from Lucas' book. /*_endCmd /********************* >>>>>>>>> Lucas's vt*[cos(Aθpc(POIo(t),t)) - 1] term see Lucas p71h0.25 I have consistently been unable to come up with this term!!! See Equation (3) in the previous sub-sub-section for my current expression. /********************* >>>>>>>>> B X v field is not electrostatic in nature I do not yet have a good grasp of this. My current understanding is that while ETodv(POIo,t) DOES use an iterative process, this process does NOT feed back into "B X v" as Lucas's point (from Cullick[8], Hooper[10, and Spencer[12]) is that E does NOT arise from the induced "B X v" E fields. At present - this statement seems contradictory, as Lucas does use the TOTAL ETodv(POIo,t) to calculate the total BTodv(POIo,t) field? /********************************************** >>> II. Derivations for a POIp = POIo(t) fixed in the particle reference frame (RFp) /***************************************** >>>>>> Basic measures /********************* >>>>>>>>> Figure "Basic measures for for the particle reference frame RFp, using POIp=POIo(tx)" Run command to see $ eog "$d_images""Howell - Chapter 4 - POIo basic - cropped.png" & Run command to see $ eog "$d_images""Howell - Chapter 4 - dp[dt _ Rpcs(POIo,t)] - cropped.png" & Reminder : The [particle, observer] reference frames [(RFp),(Rfo)] have the same scaling and orientation, and at time t=0 their origins coincide, being an exact match at that time apart for the motion of (RFp) with the particle. Reminders for Chapter 4 : Vonv(PART) = constant [Particle, observer] frames of reference (RFp) & (RFo) are IDENTICAL [scale, rotation, etc] at time t=0, apart from the given that the particle's reference frame (RFp) moves with velocity v relative to RFo. Reminders for (POIp) - Point Of Interest that is FIXED in the particle reference frame (RFp) : [Rpc,Opc,Ppc] and their derived concepts, are NOT functions of time, i.e. (POIp). Their derivatives with respect to time are zero. [Roc,Ooc,Poc], their derived concepts, and their derivatives are all functions of time, i.e. (POIp(t),t) OBJECTIVES : Where convenient, express variables as functions of FIXED variables : (POIp) -> -Vodv(PART),t,Rpcv(POIp),Aθpc(POIp),Aφpc(POIp) (POIo) -> Vodv(PART),t,Rocv(POIo),Aθoc(POIo),Aφoc(POIo) /********************* >>>>>>>>> Rpcv(POIp), Aθpc(POIp), Aφpc(POIp) are constants As per the intRoduction to this section, : This section deals with (POIp) - Points Of Interest(POI) which are fixed with respect to the particle reference frame (RFp), and that move with constant relative velocity Vonv(PART) with repect to the observer. (mathH) Rpcv(POIp) = constant (endMath) (mathH) Aθpc(POIp) = constant (endMath) (mathH) Aφpc(POIp) = Aφoc(POIp(t),t) = constant (endMath) /********************* >>>>>>>>> Rpcv(POIo(t),t) /*/*$ cat >>"$p_augmented" "$d_augment""Rpcv.txt" /*+-----+ (RFp) basis Vector expression from the Galilean transformation : /% (mathH) Rpcv(POIo(t),t) = Rocv(POIo) - Vonv(PART)*t (endMath) (mathH) Rpcs(POIo(t),t) = Rocs(POIo) - Vons(PART)*t (endMath) /*Here, only t is a variable . /*+-----+ (RFo) basis Treating R_O0_pch(POIo) & RθPI2pcs(POIo) (orthogonal directions) separately : /% Rpcv(POIo(t),t) = [Rocv(POIo) - Vonv(PART)*t)*Rθ0pch(POIo) + [Rocv(POIo) - Vonv(PART)*t)*RθPI2pch(POIo) Note that [Rθ0pch(POIo), RθPI2pch(POIo)] = [Rθ0och(POIo), RθPI2och(POIo)] : = (Rocv(POIo) - Vonv(PART)*t)*Rθ0och(POIo) + (Rocv(POIo) - Vonv(PART)*t)*RθPI2och(POIo) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vonv(PART)*t*cos(Aθ0oc(POIo) - Aθoc(Vonv)) ] **Rθ0och(POIo) +[ Rocv(POIo)*sin(Aθoc(POIo)) - Vonv(PART)*t*cos(AθPI2oc(POIo) - Aθoc(Vonv)) ] *RθPI2och(POIo) BUT : cos(Aθ0oc(POIo) - Aθoc(Vonv)) = 1 (Aθ0oc(POIo) & Aθoc(Vonv) are collinear) cos(AθPI2oc(POIo) - Aθoc(Vonv)) = 0 (Aθ0oc(POIo) & Aθoc(Vonv) are perpendicular) so continuing : = [(Rocs(POIo)*cos(Aθoc(POIo)) - Vonv(PART)*t] *Rθ0och(POIo) +[ Rocv(POIo)*sin(Aθoc(POIo))] *RθPI2och(POIo) /*Summarizing component expression : (mathH) Rpcv(POIo(t),t) = [(Rocs(POIo)*cos(Aθoc(POIo)) - Vonv(PART)*t]*Rθ0och(POIo) + [Rocv(POIo)*sin(Aθoc(POIo))]*RθPI2och(POIo) (endMath) /*Only t is a variable in the above equation. /*+--+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (length). /*_endCmd /********************* >>>>>>>>> Rpcs(POIo(t),t) /*/*$ cat >>"$p_augmented" "$d_augment""Rpcs.txt" /*+-----+ (RFp) basis /%(1) Rpcs(POIp) = |Rpcv(POIo(t),t)| -> want in terms of [t,Vonv(POIo),Rpcv(POIo,t=0),RFo] From "Rpcv(POIo(t),t) " ; (1) Rpcv(POIo(t),t) = Rocv(POIo) - Vonv(PART)*t Therefore : (1) Rpcs(POIo(t),t) = |Rpcv(POIo(t),t)| = |Rocv(POIo) - Vonv(PART)*t| Summarizing : (mathH) Rpcs(POIp) = |Rocv(POIo) - Vonv(PART)*t| (endMath) /*+-----+ (RFo) basis /%(1) Rpcs(POIo(t),t) = |Rpcv(POIo(t),t)| From "Rpcv(POIo(t),t)" : (2) Rpcv(POIo(t),t) = [(Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *R_O0_och(POIo) +[ Rocs(POIo)*sin(Aθoc(POIo)) ] *RθPI2och(POIo) Subbing (2) into (1) (1) Rpcs(POIo(t),t) = |Rpcv(POIo(t),t)| = | [(Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *R_O0_och(POIo) +[ Rocs(POIo)*sin(Aθoc(POIo)) ] *RθPI2och(POIo) | given orthogonal basis : = { [(Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t]^2 +[ Rocs(POIo)*sin(Aθoc(POIo)) ]^2 }^(1/2) = { [ (Rocs(POIo)*cos(Aθoc(POIo))]^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 +[ Rocs(POIo)*sin(Aθoc(POIo))]^2 }^(1/2) = { Rocs(POIo)^2*[cos(Aθoc(POIo))^2 + sin(Aθoc(POIo))^2] - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Summarizing : (mathH) Rpcs(POIo(t),t) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) (endMath) /*+--+ Second approach (essentially the same thing) : Distance Rpcs(POIo(t),t) from the (RFp) origin : /% Rpcs(POIo(t),t)) = |Rpcv(POIo(t),t)| = |Rocs(POIo) - Vons(PART)*t| = { [Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)]^2 + [Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))^2 }^(1/2) = { [Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t]^2 + [Rocs(POIo)*sin(Aθoc(POIo))]^2 }^(1/2) = { [Rocs(POIo)*cos(Aθoc(POIo))]^2 - 2*Rocs(POIo)*cos(Aθoc(POIo)) *Vons(PART)*t + [Vons(PART)*t]^2 + [Rocs(POIo)*sin(Aθoc(POIo))]^2 }^(1/2) = { Rocs(POIo)^2 *[cos(Aθoc(POIo))^2 + sin(Aθoc(POIo))^2 ] - 2*Rocs(POIo)*cos(Aθoc(POIo)) *Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /* as sin^2 + cos^2 = 1 : /% Rpcs(POIo(t),t)) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /*Which is the same as first approach above. /*+--+ LIMIT CHECKS : Dimensional consistency : lenght = {length^2 - length*length/time*time + length^2 } = length OK, as all terms reduce to (length). /*_endCmd /********************* >>>>>>>>> sin(Aθpc(POIo(t),t)) /*+-----+ (RFp) basis From "R_OPI2_ocs(POIo) = RθPI2pcs(POIo)" : /% (1)* RθPI2pcs(POIo) = Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t)) = constant Therefore : (mathH) sin(Aθpc(POIo(t),t)) = RθPI2pcs(POIo)/Rpcs(POIo(t),t) (endMath) /*+-----+ (RFo) basis From "Rocv(POIo), Aθoc(POIo), Aφoc(POIo) are constants" : /% (mathH) Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) (endMath) So : sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) / Rpcs(POIo(t),t) = Rocs(POIo)*sin(Aθoc(POIo)) / { [Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ]^2 + (Rocs(POIo)*sin(Aθoc(POIo)))^2 }^(1/2) Summarizing : (mathH) sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) / {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) (endMath) /*+--+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (dimensionless). /********************* >>>>>>>>> cos(Aθpc(POIo(t),t)) From (4)&(2) : /% cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / Rpcs(POIo(t),t) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { [Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ]^2 + (Rocs(POIo)*sin(Aθoc(POIo)))^2 }^(1/2) Summarizing : (mathH) cos(Aθpc(POIo(t),t)) = [Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] / {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2}^(1/2) (endMath) /*+-----+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (dimensionless). /*sin^2 + cos^2 = 1 /% (5) sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) (6) cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) /*OK by inspection. /********************* >>>>>>>>> R_O0_pcs(POIo(t),t) See Figure "Basic measures for a POIo" : /*+-----+ (RFp) basis /% (mathH) Rθ0pcs(POIo(t),t) = Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) (endMath) /*LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (length). /*+-----+ (RFo) basis Again using Figure "Basic measures for a POIo" : Distance of R_O0_pcs(POIo) from the (RFp) origin in O0ch direction (i.e. along L(PART)). Note that the notation "_O0_" is a mnemonic for theta = O (capital O) = 0 (zero) radians : /% Rθ0pcs(POIo) = Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) = Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t (mathH) Rθ0pcs(POIo(t),t) = Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t (endMath) /********************* >>>>>>>>> RθPI2pcs(POIo(t),t) Distance of RθPI2pcs(POIp) = R_OPI2_ocs(POIo) from L(PART) in Ppch=Poch direction (i.e. perpendicular to L(PART)). Note that the notation "_OPI2_" is a mnemonic for theta = O (capital O) = PI/2 radians. Note that this measure is a constant for any POIo or (POIp(t),t), independent of the motion of the particle, and is therefore a convenient basis for (RFp) claculations for (POIo) (Points Of Interest fixed in the particle frame of reference). Note : as this sub-sub-sub-section was moved, original augmented equation numbering is used here (to avoid having to edit all references elsewhere in the document). This should ultimately be cleaned up throughout the document. /*+-----+ (RFp) basis /% (mathH) RθPI2pcs(POIo(t),t) = Rpcs(POIp)*sin(Aθpc(POIp)) = constant (endMath) /*+-----+ (RFo) basis From (3p) : /% 3p) RθPI2pcs(POIo(t),t) = Rpcs(POIo(t),t) *sin(Aθpc(POIo(t),t)) = Rocs(POIo) *sin(Aθoc(POIo)) ...[Rocv(POIo),Aθoc,Aφoc] don't change with t (mathH) RθPI2pcs(POIo(t),t) = Rocs(POIo)*sin(Aθoc(POIo)) = constant (endMath) /*+--+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (length). /********************* >>>>>>>>> K0, K1, K2 for use in differentiations /% (mathH)/* for use in differentiations /% K0 = 3/2*β^2*Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2 (endMath) (mathH)/* for use in differentiations /% K2 = (-1)*λ(Vons(PART))*Q(PART)*Rpcs(POIo(t),t)^(-2) (endMath) (mathH)/* not used in derivations, see "???" /% K1 = (-3)*β^2*Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t)^(-5)*Vons(PART)*t*(cos(Aθpc(POIo(t),t)) - 1) (endMath) /* where /% (mathH) f_sphereCapSurf(x) = β*Rocs(POIo)^2*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))*∂[∂(t): x]] (endMath) /********************* >>>>>>>>> K0(t=0), K1(t=0), K2(t=0) in terms of E0ods(POIo,t) for relativistic factor, when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! substitute E0ods(POIo,t) = Q(PART)/Rpcs(POIo(t),t)^2 given that : t=0 particle and reference frames are the same, x-axis is trajectory of particle, in direction of motion wrt RFo POIo is on the trajectory of the particle in the direction of Vons(PART) then (mathH) Rocs(POIo) = Rpcs(POIo(t),t=0) when : t=0, RFp=RFo @t=0, POIo is on the trajectory of the particle in the direction of Vons(PART) (endMath) <--???--| /% (mathH)/* when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! /% K0(t=0) = E0ods(POIo,t=0)*3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 (endMath) (mathH)/* when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! /% K2(t=0) = -E0ods(POIo,t=0)*λ(Vons(PART)) (endMath) (mathH)/* not used in derivations, see "???" /% K1(t=0) = 0 (endMath) /* derivation of K1(t=0) /% K1(t=0) = -3 *β^2*Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(-5)*Vons(PART)*t*(cos(Aθpc(POIo(t),t=0)) - 1) = -3 *β^2*Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(-5)*Vons(PART)*0*(cos(Aθpc(POIo(t),t=0)) - 1) = 0 /* IFFY /********************* >>>>>>>>> EIods(POIo,t=0,ith stage) /* Note that this is a clean definition of a not-actually-recursive process : 05Oct2019 still need to do derivations, proofs must link to lines in "Howell - math of Lucas Universal Force.txt" /% (mathH) EIods(POIo,t=0,1st stage) = K_1st + f_sphereCapSurf(EIods(POIo,t)) (endMath) (mathH) EIods(POIo,t=0,2nd stage) = K_1st + f_sphereCapSurf(EIods(POIo,t=0,1st stage) (endMath) (mathH) EIods(POIo,t=0,ith stage) = K_1st + f_sphereCapSurf(EIods(POIo,t=0,(i-1) stage)))} (endMath) /********************* >>>>>>>>> K_1st, K_2nd for use in differentiations (mathH) K_1st = K0 + K2 (endMath) (mathH)/* differentiable form /% K_1st = + Q(PART) *( 3/2 *β^2*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2 - λ(Vons(PART)) *Rpcs(POIo(t),t)^(-2) ) (endMath) /* using /% 2940:(mathL)/* HIGHLY restricted! at time t=0 This means that the [observer, particle] reference frames are exactly the same at t=0. /% Rocs(POIo) = Rpcs(POIo(t),t=0) K_1st = + 3/2*β^2*Q(PART)*Rpcs(POIo(t),t=0)^(-2)*sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART)) *Q(PART)*Rpcs(POIo(t),t=0)^(-2) = Q(PART)*Rpcs(POIo(t),t=0)^(-2) *( + 3/2 *β^2 *sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART)) *1 ) /* using /% 1205:(mathH) E0ods(POIo,t) = E0pds(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2 K_1st = E0ods(POIo,t=0) *( + 3/2 *β^2 *sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART)) *1 ) (mathH)/* HIGHLY restrictive conditions : t=0, RFo(t=0) = Rfp(t=0), Rocs(POIo) = Rpcs(POIo(t),t=0) This means that the [observer, particle] reference frames are exactly the same at t=0 (other than motion). drop as roundoff error : f_sphereCapSurf expression see "textIn - relativistic factor, intermediate symbols, restrictive conditions.txt" /% K_1st = E0pds(POIp)*3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 - E0pds(POIp)*λ(Vons(PART)) (endMath) K_2nd = β*Rocs(POIo)^2 *{ + 21/2 *β^3*Q(PART)*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-6)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t)) ] - 2 *λ(Vons(PART)) *β *Q(PART)/Rocs(POIo) *Rpcs(POIo(t),t)^(-3)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t)) *cos(Aθpc(POIo(t),t)) ] } /* using /% 2715:(mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^1] = sin(Aθpc(POIo(t),t=0))^2/2 2717:(mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^3] = sin(Aθpc(POIo(t),t=0))^4/4 K_2nd = β*Rocs(POIo)^2 *{ + 21/2 *β^3*Q(PART)*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4/4 - 2 *λ(Vons(PART)) *β *Q(PART)/Rocs(POIo) *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t=0))^2/2 } = + 21/2/4 *β^4*Q(PART) *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4 - 2/2 *λ(Vons(PART)) *β^2*Q(PART)*Rocs(POIo) *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t=0))^2 = + 21/8 *β^4*Q(PART) *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4 - λ(Vons(PART)) *1 *β^2*Q(PART)*Rocs(POIo) *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t=0))^2 (mathH) K_2nd = + 21/8 *β^4*Q(PART) *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4 - λ(Vons(PART)) *1 *β^2*Q(PART)*Rocs(POIo) *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t=0))^2 (endMath) /********************* >>>>>>>>> K_1st(t=0), K_2nd(t=0), K_3rd(t=0) in terms of E0ods(POIo,t) for relativistic factor, when [t=0, RFp=RFo @t=0], use only AFTER differentiations!!! /% ????:(mathH) K_1st = 3/2*β^2*Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t=0)^(-5)*sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART))*Q(PART)*Rpcs(POIo(t),t=0)^(-2) for use in differentiations 1049:(mathH) Rocs(POIo) = Rpcs(POIo(t),t=0) when : t=0, RFp=RFo @t=0 K_1st = 3/2*β^2*Q(PART) *Rpcs(POIo(t),t=0)^(-2)*sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART))*Q(PART)*Rpcs(POIo(t),t=0)^(-2) 1109:(mathH) E0ods(POIo,t) = E0pds(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2 (mathH)/* when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! K_1st(t=0) = E0ods(POIo,t)*3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 - E0ods(POIo,t)*λ(Vons(PART)) (endMath) ????:(mathH) K_2nd = 21/8*β^4*Q(PART)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4 - λ(Vons(PART))*β^2*Q(PART)*Rocs(POIo)*Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t=0))^2 for use in differentiations 1049:(mathH) Rocs(POIo) = Rpcs(POIo(t),t=0) when : t=0, RFp=RFo @t=0 K_2nd = 21/8*β^4*Q(PART)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4 - λ(Vons(PART))*β^2*Q(PART) *Rpcs(POIo(t),t)^(-2)*sin(Aθpc(POIo(t),t=0))^2 1109:(mathH) E0ods(POIo,t) = E0pds(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2 (mathH)/* HIGHLY restricted! [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! /% K_2nd(t=0) = E0ods(POIo,t)*21/8*β^4*Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t=0))^4 - E0ods(POIo,t)*λ(Vons(PART))*β^2*sin(Aθpc(POIo(t),t=0))^2 (endMath) /********************* >>>>>>>>> E0pdv(POIp) Gauss's Law for a single point charge, in the particle reference frame (RFp) : /% E0pdv(POIp) = Q(PART) /|Rpcv(POIp)|^2 *Rpch(POIp) = Q(PART) / Rpcs(POIp) ^2 *Rpch(POIp) (mathH) E0pdv(POIp) = Q(PART)/Rpcs(POIp)^2*Rpch(POIp) (endMath) (mathH) E0pds(POIp) = Q(PART)/Rpcs(POIp)^2 (endMath) /*Here I have not yet expressed in terms of (RFo).... /********************* >>>>>>>>> E0odv(POIo,t) /* In [Maxwell, relativity] electrodynamics, time delays are ignored!! (wrong!) Therefore, the STATIC E field is the same in RFp and RFo coordinates. It is NOT restricted to t=0 when RFp=RFo, as Rpcs(POIo(t),t) is used rather than Rocs(POIo) /% (mathH) E0odv(POIo,t) = E0pdv(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2 *Rpch(POIo(t),t) (endMath) (mathH) E0ods(POIo,t) = E0pds(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2 (endMath) /********************* >>>>>>>>> B0pdv(POIp) = B0odv(POIo) = 0 In this analysis, and implicitly in Lucas (compare pp66h0.5 Equation (4-13) with p67h0.6 Equation (4-13), it is assumed that there are NO external magnetic fields, (i.e. from other sources, independent of the particle), nor is the particle magnetic, and so B0 is a constant zero (assuming that this is the definition of B0) for all reference frames. Magnetic fields must therefore arise from the electrostatic or induced electric fields arising from the charged particle. (mathH) B0pdv(POIp) = 0 (endMath) (mathH) B0odv(POIo) = 0 (endMath) /********************* >>>>>>>>> BIpdv(POIp) = 0 ≠ BIodv(POIp(t),t) = BIodv(POIo,t) As defined here, induced magnetic fields arise from the movement of electric fields, for example as arising from a moving charged particle. In this work, only field arising from the particle and its movement are considered. However, the charged particle does NOT move with respect to the particle reference frame RFp, so its relative velocity is zero, therefore NO magnetic field arises in RFp. Therefore, there is no induced magnetic field for Points Of Interest (POIp) that are fixed in the particle reference frame (RFp). This is shown in the derivation below. Generalized Ampere's Law, in (RFp) ("X" is the vector cross-product) : /% BIpdv(POIp) = Vpnv(POIp)/c X E0pdv(Rpcv(POIp)) = 0 /c X E0pdv(Rpcv(POIp)) = 0 (mathH) BIpdv(POIp) = 0 (endMath) /********************* >>>>>>>>> BTpdv(POIp) = 0 Assuming : /% BTpdv(POIp) = B0pdv(POIp) + BIpdv(POIp) = 0 + 0 = 0 /* where (mathH) B0pdv(POIp) = 0 as given in Chapter 4 (endMath) - magnetic field external (currents, permanent mags) in (RFp) BIpdv(POIp) - magnetic field induced by charge Q(PART), which moves in RFo but does NOT move in (RFp) /% (mathH) BTpdv(POIp) = 0 (endMath) /********************* >>>>>>>>> EIpdv(POIp) = 0 The INDUCED electric field arises from CHANGES in the total magnetic field at a point. In the particle reference frame there is no change in magnetic field at POIp, hence no induced electric field. (mathH) EIpdv(POIp) = 0 (endMath) /********************* >>>>>>>>> ETpdv(POIp) = E0pdv(POIp) /% ETpdv(POIp) = E0odv(POIo,t) + EIodv(POIo,t) = Q(PART)/Rpcs(POIp)^2*Rpch(POIp) + 0 = Q(PART)/Rpcs(POIp)^2*Rpch(POIp) (mathH) ETpdv(POIp) = Q(PART)/Rpcs(POIp)^2*Rpch(POIp) (endMath) /***************************************** >>>>>> Derivatives /********************* >>>>>>>>> Figure "Calculus for RFp, using POIp=POIo(t)" Run command to see $ eog "$d_images""Howell - Chapter 4 - POIo calculus - cropped.png" & /********************* >>>>>>>>> Reminders for Chapter 4 : Vonv(PART) = constant [Particle, observer] frames of reference (RFp) & (RFo) are IDENTICAL [scale, rotation, etc] at time t=0, apart from the given that the particle's reference frame (RFp) moves with velocity Vonv(PART) relative to RFo. Reminders for (POIo) - Point Of Interest that is FIXED in the observer reference frame (RFo) : [Roc,Ooc,Poc] and their derived concepts, are NOT functions of time, i.e. (POIp). Their derivatives are zero. [Rpc,Opc,Ppc], their related concepts, and their derivatives are all functions of time, i.e. (POIp) /********************* >>>>>>>>> OBJECTIVES : Where convenient, express variables as functions of FIXED variables : (POIp) -> -Vodv(PART),t,Rpcv(POIp),Aθpc(POIp),Aφpc(POIp) (POIo) -> Vodv(PART),t,Rocv(POIo),Aθoc(POIo),Aφoc(POIo) 28Sep2019 - nah, not so importnat! I just want answers /********************* >>>>>>>>> ∂[∂(t): Rpcv(POIp)] = ∂[∂(t): Aθpc(POIp)] = 0 As per "Rpcv(POIp), Aθpc(POIp), Ppc(POIp)" above, Rpc and Opc are constants for a given POIp, so their derivatives are zero : /% (mathH) ∂[∂(t): Rpcv(POIp)] = 0 (endMath) (mathH) ∂[∂(t): Aθpc(POIp)] = 0 (endMath) /********************* >>>>>>>>> ∂[∂(t): Rpcv(POIo(t) ] From Galilaean invariance, Eqn (1) in "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)" fixed in RFo space : /% Rpcv(POIo(t),t) = Rocv(POIo) - Vonv(PART)*t /*Meaning that for any POIo : /% ∂[∂(t): Rpcv(POIo(t),t) = ∂[∂(t): Rocv(POIo) - Vonv(PART)*t] = ∂[∂(t): Rocv(POIo)] - ∂[∂(t): Vonv(PART)*t] = 0 - Vonv(PART) (mathH) ∂[∂(t): Rpcv(POIo(t),t)] = -Vonv(PART) (endMath) /*This is expected. /*+-----+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to length/time /********************* >>>>>>>>> Figure "∂[∂(t): Rpcs(POIo(t),t) ]" Run command to see $ eog "$d_images""Howell - Chapter 4 - dp[dt _ Rpcs(POIo,t)] - cropped.png" & /*/*$ cat >>"$p_augmented" "$d_augment""d-dt Rpcs.txt" (mathH) ∂[∂(t): Rpcs(POIo(t),t)] = -Vons(PART)*cos(Aθpc(POIo(t),t)) (endMath) /*???????????????????????????? >>>>>>>>>>>> Confirmation of ∂[∂(t): Rpcs(POIo(t),t)] 12Oct2019 is this wrong? /% 1369:(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = -Vons(PART)*cos(Aθpc(POIo(t),t)) /* The work below was done independently of my previous work, to see if I : - obtaine the same answers - notice [missing issues, factors] The original checks can be found at : - "?documenht?" - ?section? /* see "$d_PROJECTS""Lucas - Universal Force/Images/Howell - Chapter 4 - dp[dt _ Rpcs(POIo,t)] - cropped.png" "Figure "∂[∂(t): Rpcs(POIo(t),t) ]"" In the figure, I have /% delta(Rpcs) = Rpcs(POIo(t),t2) - Rpcs(POIo(t),t1) = Vonv(PART)*delta(t)*cos(pi - Aθpc(POIo(t),t2)) /* This is wrong?!! It should be as also shown in the figure : /% (NYET!!) => Vonv(PART)*delta(t)*cos(Aθpc(POIo(t),t2) - Aθpc(POIo(t),t1)) /* Reminders : /% cos(pi - Aθpc(POIo(t),t)) = -cos(Aθpc(POIo(t),t)) sin(pi - Aθpc(POIo(t),t)) = -sin(Aθpc(POIo(t),t)) cos(pi/2 - Aθpc(POIo(t),t)) = ?[+,-]?*sin(Aθpc(POIo(t),t)) sin(pi/2 - Aθpc(POIo(t),t)) = ?[+,-]?*cos(Aθpc(POIo(t),t)) /* +-----+ Case 1 : the particle is "ahead" of the Point of Interest in the Observer frame (POIo) ie the particle is flying away from the POIo on a parallel track and Rpcs is GROWING /% Aθpc(POIo(t),t) >= pi/2 delta(Rpcs) = cos(Aθpc(POIo(t),t2))*(Vons(PART)*(t2 - t1)) = cos(Aθpc(POIo(t),t2))*Vons(PART)*delta(t) ∂[∂(t): Rpcs(POIo(t),t)] = limit as delta(t) -> 0 : = (-1)*cos(Aθpc(POIo(t),t))*Vons(PART) /* so this result is the same as /% 1369:(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = -Vons(PART)*cos(Aθpc(POIo(t),t)) /* Note that for Case 1, /% Aθpc(POIo(t) >= pi/2, /* so /% cos(Aθpc(POIo(t),t)) <= 0 ∂[∂(t): Rpcs(POIo(t),t)] >= 0 /* +-----+ Case 2 : POIo is ahead of particle ie the particle is flying toward the POIo on a parallel track and Rpcs is SHRINKING! /% Aθpc(POIo(t),t) <= pi/2 (-1)*delta(Rpcs) = cos(Aθpc(POIo(t),t1))*(Vons(PART)*(t2 - t1)) = cos(Aθpc(POIo(t),t1))*Vons(PART)*delta(t) ∂[∂(t): Rpcs(POIo(t),t)] = limit as delta(t) -> 0 : = (-1)*cos(Aθpc(POIo(t),t))*Vons(PART) /* this result is the same as Case 1 and : /% 1369:(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = -Vons(PART)*cos(Aθpc(POIo(t),t)) /* Note that for Case 2, /% Aθpc(POIo(t) <= pi/2, /* so /% cos(Aθpc(POIo(t),t)) >= 0 ∂[∂(t): Rpcs(POIo(t),t)] <= 0 So the originalo results are confirmed. /*_endCmd /********************* >>>>>>>>> ∂[∂(t): Rpcs(POIo(t),t)] /% ∂[∂(t): Rpcs(POIo(t),t) ] = ∂[∂(t): |Rpcv(POIo(t),t)|] Rpcs(POIo(t),t) = |Rpcv(POIo(t),t)| so ∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|] /********************* >>>>>>>>>>>> Kahan formulation for derivatives of magnitudes : d||z|| = u_T dotProd dz / ||z|| where u_T is the linear functional dual to z wrt ||...|| HERE I INTERPRET u_T = Rpcv(POIo(t),t) /% /*/*$ cat >>"$p_augmented" "$d_augment""Kahan formulation for derivatives of magnitudes.txt" ∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|] = Rpcv(POIo(t),t) dotPRod ∂[∂(t): Rpcv(POIo(t),t)] / |Rpcv(POIo(t),t)| from (1) above in "∂[∂(t): Rpcv(POIo(t),t)]" ∂[∂(t): Rpcv(POIo(t),t)] = -Vonv(PART) therefore : ∂[∂(t): Rpcs(POIo(t),t)] = Rpcv(POIo(t),t) dotPRod -Vonv(PART) / |Rpcv(POIo(t),t)| = |Rpcv(POIo(t),t)| *cos(Aθpc(POIo(t),t))*|-Vonv(PART)|/ |Rpcv(POIo(t),t)| = -Vons(PART) *cos(Aθpc(POIo(t),t)) (1) ∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|] = -Vons(PART)*cos(Aθpc(POIo(t),t)) /*CONFIRMATION : Show Figure "∂[∂(t): Rpcs(POIo(t),t)]" /%(2) |Rpcv(POIo(t),t)| = Rpcs(POIo(t),t) = { [(Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]^2 + [Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))]^2 }^(1/2) (3) ∂[∂(t): Rpcs(POIo(t),t)] = 1/2*{ [(Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]^2 + [Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) ]^2 }^(-1/2) *∂[∂(t): {(Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]^2 + [Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))]^2}] (4) ∂[∂(t): {(Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]^2 + [Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))]^2}] = ∂[∂(t): [(Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]^2] + ∂[∂(t): {Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))}^2] /*But for all t, the following are constants, as Vod(POI) is parallel to the coordinate axis [rO0och, rO0pch] : /% Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) = constant over t for any (POIo), (POIp) therefore ∂[∂(t): {(Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]^2 + [Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))]^2}] = ∂[∂(t): [Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))]^2] = 2 *[Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) ] *∂[∂(t): Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))] /*Where from "∂[∂(t): Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) ]" below : /% ∂[∂(t): Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))] = -Vons(PART) therefore (5) ∂[∂(t): {(Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]^2 + [Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))]^2}] = -2*Vons(PART) * Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) Repeating (3) ∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIp)|] = 1/2*{ [(Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]^2 + [Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) ]^2 }^(-1/2) *∂[∂(t): [(Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]^2 + [Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))]^2] Substituting (5) into (3) : = 1/2*{ [(Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]^2 + [Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) ]^2 }^(-1/2) -2*Vons(PART) * Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) = -Vons(PART) * Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) /Rpcs(POIo(t),t) *{ sin(Aθpc(POIo(t),t))]^2 + cos(Aθpc(POIo(t),t)) ]^2 }^(-1/2) But 1 = { sin(A0oc(POIp(t),t))^2 + cos(A0oc(POIp(t),t))^2 }^(-1/2) therefore : (6) ∂[∂(t): Rpcs(POIo(t),t)] = -Vons(PART)*cos(Aθpc(POIo(t),t)) /*This is the same as (1) and (expr) above, confirming that result. Actually, there is NO essential difference other than sign between the derivations for : /% ∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|] = -Vons(PART)*cos(Aθpc(POIo(t),t)) ∂[∂(t): Rocs(POIo)] = ∂[∂(t): |Rocv(POIp(t),t)|] = Vons(PART)*cos(Aθoc(POIp(t),t)) /*which surprises me a bit, but it shouldn't be a surprise!! /********************* OBJECTIVES : Where convenient, express variables as functions of FIXED variables : (POIp) -> -Vodv(PART),t,Rpcv(POIp),Aθpc(POIp),Aφpc(POIp) (POIo) -> Vodv(PART),t,Rocv(POIo),Aθoc(POIo),Aφoc(POIo) /*From "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)" /% (2) Rpcs(POIo(t),t)) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo)) *Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Therefore ∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ] = 1/2* { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-1/2) *∂[∂(t): Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 ] = 1/2* { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-1/2) *{ ∂[∂(t): Rocs(POIo)^2] + ∂[∂(t): - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t] + ∂[∂(t): [Vons(PART)*t]^2] } = 1/2* { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-1/2) *{ 0 ] - 2*Rocs(POIo)*cos(Aθoc(POIo)) *Vons(PART) + 2*Vons(PART)*t*Vons(PART) } = {- Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART) + Vons(PART)^2*t } /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo)) *Vons(PART)*t + [Vons(PART)*t]^2 }^(-1/2) = Vons(PART)*{- Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t} /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}^(-1/2) then : (mathH) ∂[∂(t): Rpcs(POIo(t),t)] = Vons(PART)*{- Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t} /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}^(-1/2) (endMath) /*+-----+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (length/time). I haven't yet shown equivalence between the "-Vons(PART)*cos(Aθpc(POIo(t),t))" result and equation (7)... See "∂[∂(t): Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]" - note that the ?correct? result as obtained by the "Alternative Verification" provides a small degree of confirmation of the results for this sub-sub-section. As t -> +- infinity : As t -> +- 0 : /%As cos(Aθoc(POIp(t),t)) -> 0 : /*endsection /*_endCmd /********************* >>>>>>>>> ∂[∂(t): Rpcs(POIo(t),t)^(-α)] /% ∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|] = (-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) /* example /% ∂[∂(t): Rpcs(POIo(t),t)^(-3)] = (-3)*Rpcs(POIo(t),t)^(-4)*∂[∂(t): Rpcs(POIo(t),t)] = (-3)*Rpcs(POIo(t),t)^(-4)*(-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) = 4*Vons(PART)*Rpcs(POIo(t),t)^(-4)*cos(Aθpc(POIo(t),t)) /* pattern /% ∂[∂(t): Rpcs(POIo(t),t)^(-α)] = (-α)*Rpcs(POIo(t),t)^(-α - 1)*∂[∂(t): Rpcs(POIo(t),t)] = (-α)*Rpcs(POIo(t),t)^(-α - 1)*(-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) (mathH) ∂[∂(t): Rpcs(POIo(t),t)^(-α)] = α*Vons(PART)*Rpcs(POIo(t),t)^(-α - 1)*cos(Aθpc(POIo(t),t)) (endMath) /********************* >>>>>>>>> Figure "∂[∂(t): Aθpc(POIo(t),t)]" Run command to see $ eog "$d_images""Howell - Chapter 4 - dp[dt _ Rpcs(POIo,t)] - cropped.png" & /********************* >>>>>>>>> ∂[∂(t): Aθpc(POIo(t),t)] /*/*$ cat >>"$p_augmented" "$d_augment""d-dt Aθpc.txt" see Figure "∂[∂(t): Aθpc(POIo(t),t)]" For small angles d[theta] : (1) chord = radius*d[theta] = Rpcs(POIo(t),t)*d[Aθpc(POIo(t),t)] But also from the Figure : chord = Vons(PART)*d[t]*cos(Aθpc(POIo(t),t) + PI/2) ...where d[theta] is a differential change in O But : cos(alpha) = sin(alpha + PI/2) so : (2) chord = Vons(PART)*d[t]*sin(Aθpc(POIo(t),t)) ...where d[theta] is a differential change in O Equating (5) & (6) : Rpcs(POIo(t),t)*d[Aθpc(POIo(t),t)] = Vons(PART)*d[t]*sin(Aθpc(POIo(t),t)) or : (mathH) ∂[∂(t): Aθpc(POIo(t),t)] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) (endMath) /*+-----+ Reminders for Chapter 4 : OBJECTIVES : Where convenient, express variables as functions of FIXED variables : (POIp) -> -Vodv(PART),t,Rpcv(POIp),Aθpc(POIp),Aφpc(POIp) (POIo) -> Vodv(PART),t,Rocv(POIo),Aθoc(POIo),Aφoc(POIo) /*From "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)" : /% (2) Rpcs(POIo(t),t)) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) and (5) sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (2)* & (5)* into (3) : ∂[∂(t): Aθpc(POIo(t),t)] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) = Vons(PART) *Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Finally : (mathH) ∂[∂(t): Aθpc(POIo(t),t)] = Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) / {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}^(1/2) (endMath) /*-----+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (radians/time) /%As Aθpc(POIo(t),t) -> 0 : L1) ∂[∂(t): Aθpc(POIo(t),t)] -> Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) -> Vons(PART)*Rocs(POIo)*0 /{ Rocs(POIo)^2 - 2*Rocs(POIo)*1 *Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) -> 0 /*This is OK, as when the particle is far beyond the coordinate center (t is very large), eventually the rate of anular changes go to zero. /%As Aθpc(POIo(t),t) -> PI : L2) ∂[∂(t): Aθpc(POIo(t),t)] -> Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) -> Vons(PART)*Rocs(POIo)*0 /{ Rocs(POIo)^2 - 2*Rocs(POIo)*(-1) *Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) -> 0 /*This is OK, as when the particle is far behind of the coordinate center (t is large negative), eventually the rate of anular changes go to zero. /%As Aθpc(POIo(t),t) -> PI/2 : L3a) ∂[∂(t): Aθpc(POIo(t),t)] -> Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) -> Vons(PART)*Rocs(POIo)*1 /{ Rocs(POIo)^2 - 2*Rocs(POIo)*0 *Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) -> Vons(PART)*Rocs(POIo) /{ Rocs(POIo)^2 + [Vons(PART)*t]^2 }^(1/2) Check : L3b) see Figure "Calculus for a POIo" ∂[∂(t): Aθpc(POIo(t),t)] = Vons(PART) / Rpcs(POIo(t),t) = Vons(PART) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) = Vons(PART) / { Rocs(POIo)^2 - 2*Rocs(POIo)*0 *Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) = Vons(PART) / { Rocs(POIo)^2 + [Vons(PART)*t]^2 }^(1/2) /*WRONG!! - missing Rocs(POIo) in the numerator /*_endCmd /********************* >>>>>>>>> Figure "∂[∂(t): Rpch(POIo(t),t)]" http://www.BillHowell.ca/ /media/bill/SWAPPER/Lucas - Universal Force/Images/Howell - Chapter 4 - dp[dt _ Rpch(POIo,t)] - cropped.png /********************* >>>>>>>>> ∂[∂(t): Rpch(POIo(t),t) ] see Figure "∂[∂(t): Rpch(POIo(t),t)]" Rpch(POIo(t),t) = vector[length = 1, theta = Aθpc(POIo(t),t), phi = Aφpc(POIo) = constant ] So : (1) d[Rpch(POIo(t),t)] = d[Aθpc(POIo(t),t)]*1*RDEpdh(POIo(t),t) where RDEpdh(POIo(t),t) is anchored at end of Rpch(POIo(t),t) and is at angle Aθpc(POIo(t),t) + PI/2, ie perpendicular to Rpch(POIo(t),t) angle Aφpc(POIo(t),t) doesn't change /*+-----+ (RFp) basis From "∂[∂(t): Aθpc(POIo(t),t)]" : (3) ∂[∂(t): Aθpc(POIo(t),t)] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) Subbing (3)* into (1) : (1) ∂[∂(t): Rpch(POIo(t),t)] = ∂[∂(t): Aθpc(POIo(t),t)]*1*RDEpdh(POIo(t),t) = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*RDEpdh(POIo(t),t) Summarizing : (mathH) ∂[∂(t): Rpch(POIo(t),t)] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*RDEpdh(POIo(t),t) (endMath) where RDEpdh(POIo(t),t) is anchored at end of Rpch(POIo(t),t) and is at angle Aθpc(POIo(t),t) + PI/2, ie perpendicular to Rpch(POIo(t),t) 23Jan2017 changed to PLUS PI/2? (was that way in half of the expressions - maybe should be for RFp? angle Aφpc(POIo(t),t) doesn't change /*+-----+ (RFo) basis From "Rpcs(POIo(t),t)" : (3) Rpcs(POIo(t),t) = |Rpcv(POIo(t),t)| = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) From "sin(Aθpc(POIo(t),t))" : (5) sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (3)* & (5)* into (2) : (2) ∂[∂(t): Rpch(POIo(t),t)] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*RDEpdh(POIo(t),t) = Vons(PART)*RDEpdh(POIo(t),t) *sin(Aθpc(POIo(t),t)) /Rpcs(POIo(t),t) = Vons(PART)*RDEpdh(POIo(t),t) * Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Finally : (mathH)/* where RDEpdh(POIo(t),t) is anchored at end of Rpch(POIo(t),t), is at angle Aθpc(POIo(t),t) + PI/2, ie perpendicular to Rpch(POIo(t),t), angle Aφpc(POIo(t),t) doesn't change /% ∂[∂(t): Rpch(POIo(t),t)] = Vons(PART)*RDEpdh(POIo(t),t)*Rocs(POIo)*sin(Aθoc(POIo)) / {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2} (endMath) /*+-----+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to 1/time (angles & unit vectors are dimensionless) /********************* >>>>>>>>> ∂[∂(t): sin(Aθpc(POIo(t),t))] /*MUCH LATER : use (4-17) 'Sperical coordinate transforms ' WARNING : Should do vector differentiation THEN take the magnitude!?? NOTE : Equation numbering is "weird", as I first did the "RFo" approach, whereas usually I start with the RFp approach. In consequence RFp equations start with the number "10". /*/*$ cat >>"$p_augmented" "$d_augment""d-dt sin.txt" /*+-----+ (RFp) basis see Figure "Basic measures for a POIo" : From "sin(Aθpc(POIo(t),t))" : /% (1)* sin(Aθpc(POIo(t),t)) = RθPI2pcs(POIo)/Rpcs(POIo(t),t) Therefore : ∂[∂(t): sin(Aθpc(POIo(t),t))] = ∂[∂(t): RθPI2pcs(POIo)/Rpcs(POIo(t),t)] = RθPI2pcs(POIo)*∂[∂(t): 1/Rpcs(POIo(t),t)] (1) = RθPI2pcs(POIo)*(-1)/Rpcs(POIo(t),t)^2*∂[∂(t): Rpcs(POIo(t),t)] From "∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|]" : (1)** ∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|] = -Vons(PART)*cos(Aθpc(POIo(t),t)) Subbing (1)* into (1) : (1) ∂[∂(t): sin(Aθpc(POIo(t),t))] = RθPI2pcs(POIo)*(-1)/Rpcs(POIo(t),t)^2*∂[∂(t): Rpcs(POIo(t),t)] = RθPI2pcs(POIo)*(-1)/Rpcs(POIo(t),t)^2*-Vons(PART)*cos(Aθpc(POIo(t),t)) = Vons(PART)*cos(Aθpc(POIo(t),t))*RθPI2pcs(POIo)/Rpcs(POIo(t),t)^2 But AGAIN using "sin(Aθpc(POIo(t),t))" : (1)* sin(Aθpc(POIo(t),t)) = RθPI2pcs(POIo)/Rpcs(POIo(t),t) Summarizing : (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) (endMath) /* Double-check - /% ∂[∂(t): sin(Aθpc(POIo(t),t))] = ∂[∂(Aθpc(POIo(t),t)): sin(Aθpc(POIo(t),t))] *∂[∂(t): Aθpc(POIo(t),t)] /* from above : /% ∂[∂(t): Aθpc(POIo(t),t)] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) /* therefore : /% ∂[∂(t): sin(Aθpc(POIo(t),t))] = cos(Aθpc(POIo(t),t))*Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) /* OK - this is the same as above!!! /*+-----+ (RFo) basis see Figure "Basic measures for a POIo" : /% Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) sin(Aθpc(POIo(t),t)) = Rocs(POIo)/Rpcs(POIo(t),t)*sin(Aθoc(POIo)) but [Rocs(POIo),sin(Aθoc(POIo))] are constants for (POIo), therefore ∂[∂(t): sin(Aθpc(POIo(t),t))] = Rocs(POIo)*sin(Aθoc(POIo)) * ∂[∂(t): 1/Rpcs(POIo(t),t)] = Rocs(POIo)*sin(Aθoc(POIo)) *(-1)/Rpcs(POIo(t),t)^2 *∂[∂(t): Rpcs(POIo(t),t)] from "∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|]" : (7) ∂[∂(t): Rpcs(POIo(t),t)] = [ - Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART) + Vons(PART)^2*t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART) *t)^2 ]^(1/2) /*From "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)" : /% (2) Rpcs(POIo(t),t)) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART) *t]^2 }^(1/2) therefore : ∂[∂(t): sin(Aθpc(POIo(t),t))] = Rocs(POIo)*sin(Aθoc(POIo)) *(-1)/Rpcs(POIo(t),t)^2 *∂[∂(t): Rpcs(POIo(t),t)] = -1*Rocs(POIo)*sin(Aθoc(POIo)) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART) *t]^2 }^(1/2)^2 * [ - Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART) + Vons(PART)^2*t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART) *t)^2 ]^(1/2) = -1*Rocs(POIo)*sin(Aθoc(POIo)) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART) *t]^2 } * [ - Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART) + Vons(PART)^2*t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART) *t)^2 ]^(1/2) = -1*Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART) *t]^2 } * [ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] / [ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART) *t)^2 ]^(1/2) Finally : (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))] = -Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART)*{-Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t} / {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2}^(3/2) (endMath) /*+-----+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (1/time). The result here looks a bit opaque, and this is not a good sign... See "∂[∂(t): Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]" - note that the ?correct? result as obtained by the "Alternative Verification" provides a small degree of confirmation of the results for this usb-sub-section. More detailed checks via alternate "routes" : later ... /*_endCmd /********************* >>>>>>>>> ∂[∂(t): cos(Aθpc(POIo(t),t))] Reminders for Chapter 4 : Vonv(PART) = constant [Particle, observer] frames of reference (RFp) & (RFo) are IDENTICAL [scale, rotation, etc] at time t=0, apart from the given that the particle's reference frame (RFp) moves with velocity Vonv(PART) relative to RFo. Reminders for (POIo) - Point Of Interest that is FIXED in the observer reference frame (RFo) : [Roc,Ooc,Poc] and their derived concepts, are NOT functions of time, i.e. (POIp). Their derivatives with respect to time are zero. [Rpc,Opc,Ppc], their related concepts, and their derivatives are all functions of time, i.e. (POIp) /*/*$ cat >>"$p_augmented" "$d_augment""d-dt cos.txt" /*+-----+ (RFp) basis see Figure "Basic measures for a POIo" : From "cos(Aθpc(POIo(t),t))" : /% (1)* cos(Aθpc(POIo(t),t)) = Rθ0pcs(POIo)/Rpcs(POIo(t),t) Therefore : ∂[∂(t): cos(Aθpc(POIo(t),t))] = ∂[∂(t): Rθ0pcs(POIo)/Rpcs(POIo(t),t)] = + ∂[∂(t): Rθ0pcs(POIo)] /Rpcs(POIo(t),t) + Rθ0pcs(POIo)*∂[∂(t): Rpcs(POIo(t),t)^(-1)] = + ∂[∂(t): Rθ0pcs(POIo)] *Rpcs(POIo(t),t)^(-1) + Rθ0pcs(POIo) *(-1)*Rpcs(POIo(t),t)^(-2)*∂[∂(t): Rpcs(POIo(t),t)] /* from above and Figure "Basic measures for a POIo" : /% ∂[∂(t): Rθ0pcs(POIo)] = -Vons(PART) ∂[∂(t): Rpcs(POIo(t),t)] = -Vons(PART)*cos(Aθpc(POIo(t),t)) /* therefore /% ∂[∂(t): cos(Aθpc(POIo(t),t))] = + -Vons(PART) *Rpcs(POIo(t),t)^(-1) + Rθ0pcs(POIo) *(-1)*Rpcs(POIo(t),t)^(-2)*-Vons(PART)*cos(Aθpc(POIo(t),t)) = - Vons(PART)*Rpcs(POIo(t),t)^(-1) + Vons(PART)*Rpcs(POIo(t),t)^(-2)*Rθ0pcs(POIo)*cos(Aθpc(POIo(t),t)) /* but /% Rθ0pcs(POIo) = Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) /* so /% ∂[∂(t): cos(Aθpc(POIo(t),t))] = - Vons(PART)*Rpcs(POIo(t),t)^(-1) + Vons(PART)*Rpcs(POIo(t),t)^(-2)*Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t)) = - Vons(PART)*Rpcs(POIo(t),t)^(-1) + Vons(PART)*Rpcs(POIo(t),t)^(-1)*cos(Aθpc(POIo(t),t))^2 = - Vons(PART)*Rpcs(POIo(t),t)^(-1)*(1 - cos(Aθpc(POIo(t),t))^2) = - Vons(PART)*Rpcs(POIo(t),t)^(-1)*sin(Aθpc(POIo(t),t))^2 Summarizing : (mathH) ∂[∂(t): cos(Aθpc(POIo(t),t))] = -Vons(PART)*sin(Aθpc(POIo(t),t))^2/Rpcs(POIo(t),t) (endMath) /* Double-check - /% ∂[∂(t): cos(Aθpc(POIo(t),t))] = ∂[∂(Aθpc(POIo(t),t)): cos(Aθpc(POIo(t),t))] *∂[∂(t): Aθpc(POIo(t),t)] /* from above : /% ∂[∂(t): Aθpc(POIo(t),t)] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) /* therefore : /% ∂[∂(t): cos(Aθpc(POIo(t),t))] = -sin(Aθpc(POIo(t),t))*Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) = -Vons(PART)*sin(Aθpc(POIo(t),t))^2/Rpcs(POIo(t),t) /* OK - this is the same as above!!! /*+-----+ (RFo) basis WARNING : Should do vector differentiation THEN take the magnitude!?? From "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)" 28Sep2019 - this needs to be checked!!! /% (6) cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) Therfore : ∂[∂(t): cos(Aθpc(POIo(t),t))] = ∂[∂(t): [Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) = ∂[∂(t): Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) + [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] *∂[∂(t): {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}^( - 1/2)] = -Vons(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) + [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] *(-1/2)*{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(-3/2) *∂[∂(t): Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2] = -Vons(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) + [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] *(-1/2)/{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(3/2) *{ - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART) + Vons(PART)^2*2*t } = -Vons(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) + [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] *{ Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART) - Vons(PART)^2*t } /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(3/2) = -Vons(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) + [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] * Vons(PART) *{ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t } /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(3/2) = -Vons(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) + Vons(PART) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] *{ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t } /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(3/2) = -Vons(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) + Vons(PART) *{ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t }^2 /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(3/2) = Vons(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) *{ -1 + { Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t }^2 /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 } } Finally, (1) ∂[∂(t): cos(Aθpc(POIo(t),t))] = Vons(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) *{ -1 + { Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t }^2 /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 } } /*+-----+ LIMIT CHECKS : later ... Dimensional consistency - OK, as all terms reduce to (1/time). /*_endCmd /********************* >>>>>>>>> ∂[∂(t): Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))] = 0 WARNING : Should do vector differentiation THEN take the magnitude!?? Looking at Figure "∂[∂(t): Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]", it is clear by inspection that : /% /*/*$ cat >>"$p_augmented" "$d_augment""Rpcs*sin.txt" (mathH) ∂[∂(t): Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))] = 0 (endMath) (mathH) ∂[∂(t): ROPI2pcs(POIo(t),t)] = 0 (endMath) /*+-----+ (RFp) basis see Figure "Basic measures for a POIo" : ALTERNATIVE DERIVATION This also serves to test the constituent relations. /% ∂[∂(t): Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))] = + ∂[∂(t): Rpcs(POIo(t),t)] *sin(Aθpc(POIo(t),t)) + Rpcs(POIo(t),t) *∂[∂(t): sin(Aθpc(POIo(t),t))] /* from above /% ∂[∂(t): Rpcs(POIo(t),t)] = -Vons(PART)*cos(Aθpc(POIo(t),t)) ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) /* substituting /% ∂[∂(t): Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))] = + -Vons(PART)*cos(Aθpc(POIo(t),t)) *sin(Aθpc(POIo(t),t)) + Rpcs(POIo(t),t) *Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) = - Vons(PART)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)) + Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*Rpcs(POIo(t),t) = Vons(PART)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))*(-1 + 1) = 0 ∂[∂(t): Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))] = 0 /*+-----+ (RFo) basis see Figure "Basic measures for a POIo" : ALTERNATIVE DERIVATION This also serves to test the constituent relations. /% ∂[∂(t): Rpcs(POIo(t),t) *sin(Aθpc(POIo(t),t))] = ∂[∂(t): Rpcs(POIo(t),t)]*sin(Aθpc(POIo(t),t)) + Rpcs(POIo(t),t)*∂[∂(t): sin(Aθpc(POIo(t),t))] from "∂[∂(t): Rpcs(POIo(t),t) ] = ∂[∂(t): |Roc(POIo)|]" : (7) ∂[∂(t): Rpcs(POIo(t),t)] = [ - Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART) + Vons(PART)^2*t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART) *t)^2 ]^(1/2) from "∂[∂(t): sin(Aθpc(POIo(t),t)) ]" : (1) ∂[∂(t): sin(Aθpc(POIo(t),t))] = -Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) * [ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] / [ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART) *t)^2 ]^(3/2) so ∂[∂(t): Rpcs(POIo(t),t) *sin(Aθpc(POIo(t),t))] = ∂[∂(t): Rpcs(POIo(t),t)]*sin(Aθpc(POIo(t),t)) + Rpcs(POIo(t),t)*∂[∂(t): sin(Aθpc(POIo(t),t))] = sin(Aθpc(POIo(t),t)) * [ - Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART) + Vons(PART)^2*t ] / [ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART) *t)^2 ]^(1/2) + Rpcs(POIo(t),t) * -Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) * [ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] / [ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART) *t)^2 ]^(3/2) /*From "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)" : /% (2) Rpcs(POIo(t),t)) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /*From "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)" : /% (5) sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) so ∂[∂(t): Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))] = Rocs(POIo)*sin(Aθoc(POIo)) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART) *t]^2 }^(1/2) * [ - Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART) + Vons(PART)^2*t ] / [ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART) *t)^2 ]^(1/2) + { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART) *t]^2 }^(1/2) * -Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) * [ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] / [ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART) *t)^2 ]^(3/2) = Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) * [ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] / [ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART) *t)^2 ] + -Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) * [ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] * { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART) *t]^2 }^(1/2) / [ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART) *t)^2 ]^(3/2) = Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) * [ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] / [ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART) *t)^2 ] - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) * [ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] / [ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART) *t)^2 ] = [ Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) ] * [ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] / [ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART) *t)^2 ] = 0 * [ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] / [ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART) *t)^2 ] = 0 /* Finally : /% ∂[∂(t): Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))] = 0 /* OK, it is the same /*This makes sense, as from Figure "Basic measures for a POIo", Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t)) is a constant because the endpoint of rov is always on L(POI), parallel to L(PART), the trajectory of the center of the particle. /% Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo) /*+-----+ LIMIT CHECKS : Dimensional consistency - OK in intermediate equations, as all terms reduce to (length/time). The agreement between the [geometric, algebraic] approaches above provides a small but essential degree of confirmation of the result. /*_endCmd /********************* >>>>>>>>> ∂[∂(t): Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))] WARNING : Should do vector differentiation THEN take the magnitude!?? Looking at Figure "Basic measures for a POIo", it is clear by inspection that : /% /*/*$ cat >>"$p_augmented" "$d_augment""d-dt Rpcs*_cos.txt" (mathH) ∂[∂(t): Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))] = -Vons(PART) (endMath) (mathH) ∂[∂(t): Rθ0pcs(POIo(t),t)] = -Vons(PART) (endMath) /*+-----+ ALTERNATIVE DERIVATION This also serves to test the constituent relations. /%(2) ∂[∂(t): Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))] = ∂[∂(t): Rpcs(POIo(t),t)] *cos(Aθpc(POIo(t),t)) + Rpcs(POIo(t),t) *∂[∂(t): cos(Aθpc(POIo(t),t))] /*+-----+ FIRST TERM /%from "∂[∂(t): Rpcs(POIo(t),t) ] = ∂[∂(t): |Roc(POIo)|]" : (7) ∂[∂(t): Rpcs(POIo(t),t)] = Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(1/2) /*From "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)" : /% (6) cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) So : (3) ∂[∂(t): Rpcs(POIo(t),t)] *cos(Aθpc(POIo(t),t)) = Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(1/2) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) = -Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 } = -Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ]^2 /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 } /*+-----+ SECOND TERM /*From "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)" : /% (2) Rpcs(POIo(t),t)) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) from "∂[∂(t): cos(Aθpc(POIo(t),t)) ]" : (1) ∂[∂(t): cos(Aθpc(POIo(t),t)) ] = Vons(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) *{ -1 + { Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t }^2 /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 } } So : (4) Rpcs(POIo(t),t)*∂[∂(t): cos(Aθpc(POIo(t),t))] = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) * Vons(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) *{ -1 + { Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t }^2 /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 } } = Vons(PART) *{ -1 + { Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t }^2 /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 } } /*+-----+ COMBINATION of first & second terms : Subbing (3)&(4) into (2) : (2) ∂[∂(t): Rpcs(POIo(t),t) *cos(Aθpc(POIo(t),t))] = ∂[∂(t): Rpcs(POIo(t),t)] *cos(Aθpc(POIo(t),t)) + Rpcs(POIo(t),t) *∂[∂(t): cos(Aθpc(POIo(t),t))] ={ -Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ]^2 /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 } } + { Vons(PART) *{ -1 + { Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t }^2 /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 } } } = Vons(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 } *{-[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ]^2 +{ -1 *{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 } + { Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t }^2 } } = Vons(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 } *{-[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ]^2 +{ -1 *{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 } + { Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t }^2 } = Vons(PART) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 } *{-[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ]^2 +{ -1 *{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 } +[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t }^2 } = -Vons(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 } *{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 } = -Vons(PART) Finally : (5) ∂[∂(t): Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))] = -Vons(PART) /*Which is the same as (1) above. /*+-----+ LIMIT CHECKS : Dimensional consistency - OK in intermediate equations, as all terms reduce to (length/time). The agreement between the [geometric, algebraic] approaches above provides a small but essential degree of confirmation of the result. /*_endCmd /********************* >>>>>>>>> ∂[∂(t): Rpcs(POIo(t),t)^(-b)*sin(Aθpc(POIo(t),t))^a] 28Sep2019 - just started fixing for ∂[∂(t): Aθpc(POIo(t),t)] this is a HUGE CHANGE, much simpler than all previous work!!! to generate (mathH) lines : see "Binomial Series for Chapter 4.ods" /*/*$ cat >>"$p_augmented" "$d_augment""d-dt Rpcs^-b*sin^a.txt" /* General pattern from general case, and checked with specific examples below : /% (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^a*Rpcs(POIo(t),t)^(-β)] = (a+β)*Vons(PART)*sin(Aθpc(POIo(t),t))^a*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-β - 1) (endMath) Use full screen mode to more easily see the equations. (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^1] = 1*Vons(PART)*sin(Aθpc(POIo(t),t))^1*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-1) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^2] = 2*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-1) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^3] = 3*Vons(PART)*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-1) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^4] = 4*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-1) (endMath) (mathH) ∂[∂(t): Rpcs(POIo(t),t)^( 1)] = (-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) (endMath) (mathH) ∂[∂(t): Rpcs(POIo(t),t)^(-1)] = Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-2) (endMath) (mathH) ∂[∂(t): Rpcs(POIo(t),t)^(-2)] = 2*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) (endMath) (mathH) ∂[∂(t): Rpcs(POIo(t),t)^(-3)] = 3*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^1*Rpcs(POIo(t),t)] = 0 (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^1*Rpcs(POIo(t),t)^(-3)] = 4*Vons(PART)*sin(Aθpc(POIo(t),t))^1*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^1*Rpcs(POIo(t),t)^(-4)] = 5*Vons(PART)*sin(Aθpc(POIo(t),t))^1*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^1*Rpcs(POIo(t),t)^(-5)] = 6*Vons(PART)*sin(Aθpc(POIo(t),t))^1*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-2)] = 4*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-3)] = 5*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-4)] = 6*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-5)] = 7*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^3*Rpcs(POIo(t),t)^(-3)] = 6*Vons(PART)*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^3*Rpcs(POIo(t),t)^(-4)] = 7*Vons(PART)*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^3*Rpcs(POIo(t),t)^(-6)] = 9*Vons(PART)*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-2)] = 6*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-4)] = 8*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-5)] = 9*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-6)] = 10*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-7)] = 11*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-8) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^5*Rpcs(POIo(t),t)^(-3)] = 8*Vons(PART)*sin(Aθpc(POIo(t),t))^5*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^5*Rpcs(POIo(t),t)^(-5)] = 10*Vons(PART)*sin(Aθpc(POIo(t),t))^5*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^5*Rpcs(POIo(t),t)^(-7)] = 12*Vons(PART)*sin(Aθpc(POIo(t),t))^5*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-8) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^5*Rpcs(POIo(t),t)^(-8)] = 13*Vons(PART)*sin(Aθpc(POIo(t),t))^5*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-9) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^6*Rpcs(POIo(t),t)^(-1)] = 7*Vons(PART)*sin(Aθpc(POIo(t),t))^6*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-2) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^6*Rpcs(POIo(t),t)^(-4)] = 10*Vons(PART)*sin(Aθpc(POIo(t),t))^6*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^6*Rpcs(POIo(t),t)^(-6)] = 12*Vons(PART)*sin(Aθpc(POIo(t),t))^6*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^6*Rpcs(POIo(t),t)^(-7)] = 13*Vons(PART)*sin(Aθpc(POIo(t),t))^6*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-8) (endMath) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^6*Rpcs(POIo(t),t)^(-8)] = 14*Vons(PART)*sin(Aθpc(POIo(t),t))^6*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-9) (endMath) Derivations : /* check ∂[∂(t): sin(Aθpc(POIo(t),t))] in 2 different ways : /% 1537:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) 1351:(mathH) ∂[∂(t): Aθpc(POIo(t),t)] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) ∂[∂(t): sin(Aθpc(POIo(t),t))] = ∂[∂(Aθpc(POIo(t),t)): sin(Aθpc(POIo(t),t))] * ∂[∂(t): Aθpc(POIo(t),t)] = cos(Aθpc(POIo(t),t))] * ∂[∂(t): Aθpc(POIo(t),t)] = cos(Aθpc(POIo(t),t))] * Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))]/Rpcs(POIo(t),t) OK -> The results are the same... /* +-----+ /* Looking at "∂[∂(t): sin(Aθpc(POIo(t),t))^a*Rpcs(POIo(t),t)^(-β)]" /% ∂[∂(t): sin(Aθpc(POIo(t),t))^a *Rpcs(POIo(t),t)^(-β)] = ∂[∂(t): sin(Aθpc(POIo(t),t))^a] *Rpcs(POIo(t),t)^(-β) + sin(Aθpc(POIo(t),t))^a *∂[∂(t): Rpcs(POIo(t),t)^(-β)] = a*sin(Aθpc(POIo(t),t))^(a-1) *∂[∂(t): sin(Aθpc(POIo(t),t))] *Rpcs(POIo(t),t)^(-β) + sin(Aθpc(POIo(t),t))^a *(-β)*Rpcs(POIo(t),t)^(-β - 1)*∂[∂(t): Rpcs(POIo(t),t)] /* using /% 1132:(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = (-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) 1537:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) ∂[∂(t): sin(Aθpc(POIo(t),t))^a *Rpcs(POIo(t),t)^(-β)] = a*sin(Aθpc(POIo(t),t))^(a-1)*Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) *Rpcs(POIo(t),t)^(-β) + sin(Aθpc(POIo(t),t))^a*(-β)*Rpcs(POIo(t),t)^(-β - 1)*(-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) = a*Vons(PART)*sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a-1)*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) *Rpcs(POIo(t),t)^(-β) + (-β)*(-1)*Vons(PART)*sin(Aθpc(POIo(t),t))^a*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-β - 1) = a*Vons(PART)*sin(Aθpc(POIo(t),t))^a*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-β - 1) + β*Vons(PART)*sin(Aθpc(POIo(t),t))^a*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-β - 1) = (a+β)*Vons(PART)*sin(Aθpc(POIo(t),t))^a*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-β - 1) /* double checks with examples used during early stages of derivations /* +-----+ /* Looking at "∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-2)]" /% ∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^( - 2)] = ∂[∂(t): sin(Aθpc(POIo(t),t))^2] *Rpcs(POIo(t),t)^(-2) + sin(Aθpc(POIo(t),t))^2*∂[∂(t): Rpcs(POIo(t),t)^( - 2)] = 2*sin(Aθpc(POIo(t),t))^1*∂[∂(t): sin(Aθpc(POIo(t),t))] *Rpcs(POIo(t),t)^(-2) + sin(Aθpc(POIo(t),t))^2*(-2)*Rpcs(POIo(t),t)^(-3)*∂[∂(t): Rpcs(POIo(t),t)] /* using /% 1132:(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = (-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) 1537:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) ∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^( - 2)] = 2*sin(Aθpc(POIo(t),t))^1*Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*Rpcs(POIo(t),t)^(-2) + sin(Aθpc(POIo(t),t))^2*(-2)*Rpcs(POIo(t),t)^(-3)*-Vons(PART)*cos(Aθpc(POIo(t),t)) = 2*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) + 2*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) = 4*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) 28Sep2019 BIG CHANGE - with proper derivatives!!!!!!!!!!!!!!!!!!!!!!! /* +-----+ /* Looking at "∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-2)]" /% ∂[∂(t): sin(Aθpc(POIo(t),t))^4 *Rpcs(POIo(t),t)^(-2)] = ∂[∂(t): sin(Aθpc(POIo(t),t))^4]*Rpcs(POIo(t),t)^(-2) + sin(Aθpc(POIo(t),t))^4 *∂[∂(t): Rpcs(POIo(t),t)^(-2)] = 4*sin(Aθpc(POIo(t),t))^3*∂[∂(t): sin(Aθpc(POIo(t),t))]*Rpcs(POIo(t),t)^(-2) + sin(Aθpc(POIo(t),t))^4*(-2)*Rpcs(POIo(t),t)^(-3) *∂[∂(t): Rpcs(POIo(t),t)] /* using /% 1132:(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = (-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) 1537:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) ∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^( - 2)] = 4*sin(Aθpc(POIo(t),t))^3*Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*Rpcs(POIo(t),t)^(-2) + sin(Aθpc(POIo(t),t))^4*(-2)*Rpcs(POIo(t),t)^(-3)*-Vons(PART)*cos(Aθpc(POIo(t),t)) = 4*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) + 2*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) = 6*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) /* +-----+ /* Looking at "∂[∂(t): Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))]" /% ∂[∂(t): Rpcs(POIo(t),t)^(-5) *sin(Aθpc(POIo(t),t))] = ∂[∂(t): Rpcs(POIo(t),t)^(-5)]*sin(Aθpc(POIo(t),t)) + Rpcs(POIo(t),t)^(-5)*∂[∂(t): sin(Aθpc(POIo(t),t))] = (-5) *Rpcs(POIo(t),t)^(-6) *∂[∂(t): Rpcs(POIo(t),t)]*sin(Aθpc(POIo(t),t)) + Rpcs(POIo(t),t)^(-5)*∂[∂(t): sin(Aθpc(POIo(t),t))] /* using /% 1132:(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = (-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) 1537:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) ∂[∂(t): Rpcs(POIo(t),t)^( - 5) *sin(Aθpc(POIo(t),t))] = (-5)*Rpcs(POIo(t),t)^(-6)*-Vons(PART)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)) + Rpcs(POIo(t),t)^(-5)*Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) = (-5)*-Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) + Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) = 6*Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) /* +-----+ /* Looking at "∂[∂(t): Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2 ]" /% ∂[∂(t): Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2] = ∂[∂(t): Rpcs(POIo(t),t)^(-5)] *sin(Aθpc(POIo(t),t))^2 + Rpcs(POIo(t),t)^(-5)*∂[∂(t): sin(Aθpc(POIo(t),t))^2] = (-5) *Rpcs(POIo(t),t)^(-6)*∂[∂(t): Rpcs(POIo(t),t)]*sin(Aθpc(POIo(t),t))^2 + Rpcs(POIo(t),t)^(-5)*2*sin(Aθpc(POIo(t),t))*∂[∂(t): sin(Aθpc(POIo(t),t))] /* using /% 1132:(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = (-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) 1537:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) ∂[∂(t): Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2] = - 5*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^2 *∂[∂(t): Rpcs(POIo(t),t)] + 2*Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t)) *∂[∂(t): sin(Aθpc(POIo(t),t))] = - 5*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^2 *(-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) + 2*Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t)) * Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) = + 5*Vons(PART)*sin(Aθpc(POIo(t),t))^2 *cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) + 2*Vons(PART)*sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*Rpcs(POIo(t),t)^(-5) = + 5*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) + 2*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) = 7*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) /* +-----+ /* Looking at "∂[∂(t): Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^4 ]" /% ∂[∂(t): Rpcs(POIo(t),t)^( - 5)*sin(Aθpc(POIo(t),t))^4] = ∂[∂(t): Rpcs(POIo(t),t)^( - 5)] *sin(Aθpc(POIo(t),t))^4 + Rpcs(POIo(t),t)^(-5)*∂[∂(t): sin(Aθpc(POIo(t),t))^4] = (-5) *Rpcs(POIo(t),t)^(-6)*∂[∂(t): Rpcs(POIo(t),t)] *sin(Aθpc(POIo(t),t))^4 + Rpcs(POIo(t),t)^(-5)*4*sin(Aθpc(POIo(t),t))^3*∂[∂(t): sin(Aθpc(POIo(t),t))] /* using /% 1132:(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = (-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) 1537:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) ∂[∂(t): Rpcs(POIo(t),t)^( - 5)*sin(Aθpc(POIo(t),t))^4] = (-5) *Rpcs(POIo(t),t)^(-6)*(-1)*Vons(PART)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^4 + Rpcs(POIo(t),t)^(-5)*4*sin(Aθpc(POIo(t),t))^3*Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) = (-5)*(-1) *Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) +4 *Vons(PART)*sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*Rpcs(POIo(t),t)^(-5) = + 5*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) + 4*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) = 9*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) /* +-----+ /* Looking at "∂[∂(t): Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^3]" /% ∂[∂(t): Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^3] = ∂[∂(t): Rpcs(POIo(t),t)^(-6)] *sin(Aθpc(POIo(t),t))^3 + Rpcs(POIo(t),t)^(-6) *∂[∂(t): sin(Aθpc(POIo(t),t))^3] = (-6)*Rpcs(POIo(t),t)^(-7)*∂[∂(t): Rpcs(POIo(t),t)]*sin(Aθpc(POIo(t),t))^3 + Rpcs(POIo(t),t)^(-6)*3*sin(Aθpc(POIo(t),t))^2 *∂[∂(t): sin(Aθpc(POIo(t),t))] /* using /% 1132:(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = (-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) 1537:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) ∂[∂(t): Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^3] = (-6) *Rpcs(POIo(t),t)^(-7)*(-1)*Vons(PART)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3 + Rpcs(POIo(t),t)^(-6)*3*sin(Aθpc(POIo(t),t))^2 *Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) = (-6)*(-1) *Vons(PART)*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) + 3*Vons(PART)*sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*Rpcs(POIo(t),t)^(-6) = + 6*Vons(PART)*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) + 3*Vons(PART)*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) = + 9*Vons(PART)*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) /* +-----+ /* Looking at "∂[∂(t): Rpcs(POIo(t),t)^(-7)*sin(Aθpc(POIo(t),t))^4 ]" /% ∂[∂(t): Rpcs(POIo(t),t)^(-7)*sin(Aθpc(POIo(t),t))^4] = ∂[∂(t): Rpcs(POIo(t),t)^(-7)] *sin(Aθpc(POIo(t),t))^4 + Rpcs(POIo(t),t)^(-7) *∂[∂(t): sin(Aθpc(POIo(t),t))^4] = (-7) *Rpcs(POIo(t),t)^(-8)*∂[∂(t): Rpcs(POIo(t),t)] *sin(Aθpc(POIo(t),t))^4 + Rpcs(POIo(t),t)^(-7)*4*sin(Aθpc(POIo(t),t))^3 *∂[∂(t): sin(Aθpc(POIo(t),t))] /* using /% 1132:(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = (-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) 1537:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) ∂[∂(t): Rpcs(POIo(t),t)^(-7)*sin(Aθpc(POIo(t),t))^4] = (-7) *Rpcs(POIo(t),t)^(-8)*(-1)*Vons(PART)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^4 + Rpcs(POIo(t),t)^(-7)*4*sin(Aθpc(POIo(t),t))^3*Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) = (-7)*(-1) *Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-8) + 4* Vons(PART)*sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*Rpcs(POIo(t),t)^(-7) = + 7*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-8) + 4*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-8) = 11*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-8) /*_endCmd /********************* >>>>>>>>> ∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] 24Sep2019 initial versions 16Oct2019 file_inserts for two cases /% +-----+ /*/*$ cat >>"$p_augmented" "$d_Lucas""math Howell/cos - 1 noo, iterative, non-feedback/d-dt Rpcs^-5*t*_cos - 1.txt" /* This OLD version is WRONG! but it does reflect Lucas's recommendation 16Oct2019 This version ARBITRARILY sets ∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] equal to zero, which is not actuially correct, but it does help generate the proper functional for the infinite series for E and E* THIS IS WRONG!!! - contradiction of even using sin(Aθpc(POIo(t),t))!!!! /% 1) ∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] = ∂[∂(t): Rpcs(POIo(t),t)^(-5)]*t*(cos(Aθpc(POIo(t),t)) - 1) + Rpcs(POIo(t),t)^(-5)*∂[∂(t): t]*(cos(Aθpc(POIo(t),t)) - 1) + Rpcs(POIo(t),t)^(-5)*t *∂[∂(t): (cos(Aθpc(POIo(t),t)) - 1) /* for /% ∂[∂(t): Rpcs(POIo(t),t)^(-5)] /* from above somewhere : /% ∂[∂(t): Rpcs(POIo(t),t)^(-α)] = α*Vons(PART)*Rpcs(POIo(t),t)^(-α - 1)*cos(Aθpc(POIo(t),t)) /* therefore /% a) ∂[∂(t): Rpcs(POIo(t),t)^(-5)] = 5*Vons(PART)*Rpcs(POIo(t),t)^(-6)*cos(Aθpc(POIo(t),t)) b) ∂[∂(t): t] = 1 /* for ∂[∂(t): (cos(Aθpc(POIo(t),t)) - 1) /% ∂[∂(t): (cos(Aθpc(POIo(t),t)) - 1) = ∂[∂(t): (cos(Aθpc(POIo(t),t))] c) ` ∂[∂(t): (cos(Aθpc(POIo(t),t)) - 1) = (-1)*sin(Aθpc(POIo(t),t)) /* putting (a-c) into (1) /% ∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] = ∂[∂(t): Rpcs(POIo(t),t)^(-5)]*t*(cos(Aθpc(POIo(t),t)) - 1) + Rpcs(POIo(t),t)^(-5)*∂[∂(t): t]*(cos(Aθpc(POIo(t),t)) - 1) + Rpcs(POIo(t),t)^(-5)*t*∂[∂(t): (cos(Aθpc(POIo(t),t)) - 1) = 5*Vons(PART)*Rpcs(POIo(t),t)^(-6)*cos(Aθpc(POIo(t),t))*t*(cos(Aθpc(POIo(t),t)) - 1) + Rpcs(POIo(t),t)^(-5) *(cos(Aθpc(POIo(t),t)) - 1) - Rpcs(POIo(t),t)^(-5)*t*sin(Aθpc(POIo(t),t)) /* HIGHLY restrictive conditions!!! - RFp = RFo, particle is at origin (both reference frames) - for a POIo along direction of flight of particle, so cos(Aθpc(POIo(t),t)) = 1 THIS IS WRONG!!! - it is contradiction of even using sin(Aθpc(POIo(t),t)) - t=0 /% ∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] = 5*Vons(PART)*Rpcs(POIo(t),t)^(-6)*cos(Aθpc(POIo(t),t))*0*(cos(Aθpc(POIo(t),t)) - 1) + Rpcs(POIo(t),t)^(-5) *(1 - 1) - Rpcs(POIo(t),t)^(-5)*0*sin(Aθpc(POIo(t),t)) = 0 /* the second derivative in 4-33 work file Equation (4) is ARBITRARILY set = 0 therefore K1 = 0 /*_endCmd +-----+ /*/*$ cat >>"$p_augmented" "$d_Lucas""math Howell/cos - 1 yes, iterative, non-feedback/d-dt Rpcs^-5*t*_cos - 1.txt" /home/bill/Lucas - Universal Force/individual formulae developments/d-dt Rpcs^5*t*_cos - 1.txt www.BillHowell.ca 24Sep2019 initial based on past work 15Oct2019 - WRONG!!!!!! I cannot drop : ∂[∂(t): Rpcs(POIo(t),t=0)^(-5)*(t=0)*(cos(Aθpc(POIo(t),t=0)) - 1)] = + Rpcs(POIo(t),t=0)^(-5)*(cos(Aθpc(POIo(t),t=0)) - 1) This file is used for the "correct aqs far as I can tell" derivations of equations (4-[32-37]). Cool! derivative = original expression without t at t=0 !!! /********************* 24Sep2019 >>>>>> ∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] = 0 /% 1) ∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] a) = ∂[∂(t): Rpcs(POIo(t),t)^(-5)]*t*cos(Aθpc(POIo(t),t))] b) - ∂[∂(t): Rpcs(POIo(t),t)^(-5) *t] /*+-----+ /* looking at (1a) /% 2) ∂[∂(t): Rpcs(POIo(t),t)^(-5) *t *cos(Aθpc(POIo(t),t))] a) = ∂[∂(t): Rpcs(POIo(t),t)^(-5)] *t *cos(Aθpc(POIo(t),t)) b) + Rpcs(POIo(t),t)^(-5) *∂[∂(t): t] *cos(Aθpc(POIo(t),t)) c) + Rpcs(POIo(t),t)^(-5) *t *∂[∂(t): cos(Aθpc(POIo(t),t))] /* from section "∂[∂(t): Rpcs(POIo(t),t)^(-α)]" : /% ∂[∂(t): Rpcs(POIo(t),t)^(-α)] = α*Vons(PART)*Rpcs(POIo(t),t)^(-α - 1)*cos(Aθpc(POIo(t),t)) /* therefore /% d) ∂[∂(t): Rpcs(POIo(t),t)^(-5)] = 5*Vons(PART)*Rpcs(POIo(t),t)^(-6)*cos(Aθpc(POIo(t),t)) /* noting /% e) ∂[∂(t): t] = 1 f) ∂[∂(t): cos(Aθpc(POIo(t),t))] = -sin(Aθpc(POIo(t),t))*∂[∂(t): Aθpc(POIo(t),t))] /* from "Howell - independent math for Lucas Universal Force, Chapter 4.txt" section "∂[∂(t): Aθpc(POIo(t),t))]" : /% g) ∂[∂(t): Aθpc(POIo(t),t)] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) /* therefore /% h) ∂[∂(t): cos(Aθpc(POIo(t),t))] = -sin(Aθpc(POIo(t),t))*Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) = (-1)*Vons(PART)*sin(Aθpc(POIo(t),t))^2/Rpcs(POIo(t),t) /* substitute (2d,e,h) into (2a-c) /% 3) ∂[∂(t): Rpcs(POIo(t),t)^(-5) *t *cos(Aθpc(POIo(t),t))] a) = 5*Vons(PART)*Rpcs(POIo(t),t)^(-6)*cos(Aθpc(POIo(t),t)) *t *cos(Aθpc(POIo(t),t)) b) + Rpcs(POIo(t),t)^(-5) *1 *cos(Aθpc(POIo(t),t)) c) + Rpcs(POIo(t),t)^(-5) *t *(-1)*Vons(PART)*sin(Aθpc(POIo(t),t))^2/Rpcs(POIo(t),t) /* collecting & rearranging terms /% 4) ∂[∂(t): Rpcs(POIo(t),t)^(-5) *cos(Aθpc(POIo(t),t)) *t ] a) = 5*Vons(PART)*Rpcs(POIo(t),t)^(-6)*cos(Aθpc(POIo(t),t))^(2) *t b) + Rpcs(POIo(t),t)^(-5)*cos(Aθpc(POIo(t),t)) c) - Vons(PART)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^(2) *t /*+-----+ /* looking at (1b) /% 5) ∂[∂(t): Rpcs(POIo(t),t)^(-5) *t] a) = ∂[∂(t): Rpcs(POIo(t),t)^(-5)] *t b) + Rpcs(POIo(t),t)^(-5) *∂[∂(t): t] /* sub (2d,e) into (5a,b) /% 6) ∂[∂(t): Rpcs(POIo(t),t)^(-5) *t] a) = 5*Vons(PART)*Rpcs(POIo(t),t)^(-6)*cos(Aθpc(POIo(t),t)) *t b) + Rpcs(POIo(t),t)^(-5) *1 /*+-----+ sub (4,6) into (1) /% 7) ∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] a) = + 5*Vons(PART)*Rpcs(POIo(t),t)^(-6)*cos(Aθpc(POIo(t),t))^(2) *t b) + Rpcs(POIo(t),t)^(-5)*cos(Aθpc(POIo(t),t)) c) - Vons(PART)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^(2) *t d) - 5*Vons(PART)*Rpcs(POIo(t),t)^(-6)*cos(Aθpc(POIo(t),t)) *t e) - Rpcs(POIo(t),t)^(-5) /* At time t=0 /% 8) ∂[∂(t): Rpcs(POIo(t),t=0)^(-5)*(t=0)*(cos(Aθpc(POIo(t),t=0)) - 1)] = + Rpcs(POIo(t),t=0)^(-5)* cos(Aθpc(POIo(t),t=0)) - Rpcs(POIo(t),t=0)^(-5) /* finally /% (mathH) ∂[∂(t): Rpcs(POIo(t),t=0)^(-5)*(t=0)*(cos(Aθpc(POIo(t),t=0)) - 1)] = + Rpcs(POIo(t),t=0)^(-5)*(cos(Aθpc(POIo(t),t=0)) - 1) (endMath) /*_endCmd /********************* >>>>>>>>> ∂[∂(t): K0,K2] for use in differentiations /% 1005:(mathH)# for use in differentiations K0 = 3/2*β^2*Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2 ∂[∂(t): K0] = ∂[∂(t): 3/2*β^2*Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2] = 3/2*β^2*Rocs(POIo)^3*Q(PART) *∂[∂(t): Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2] 2121:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-5)] = 7*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) = 3/2*β^2*Rocs(POIo)^3*Q(PART) *7*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) (mathH)/* for use in differentiations /% ∂[∂(t): K0] = 21/2*β^2*Rocs(POIo)^3*Q(PART)*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) (endMath) 1006:(mathH) K2 = (-1)*λ(Vons(PART))*Q(PART)*Rpcs(POIo(t),t)^(-2) for use in differentiations ∂[∂(t): K2] = ∂[∂(t): (-1)*λ(Vons(PART))*Q(PART)*Rpcs(POIo(t),t)^(-2)] = (-1)*λ(Vons(PART))*Q(PART) *∂[∂(t): Rpcs(POIo(t),t)^(-2)] 1417:(mathH) ∂[∂(t): Rpcs(POIo(t),t)^(-α)] = α*Vons(PART)*Rpcs(POIo(t),t)^(-α - 1)*cos(Aθpc(POIo(t),t)) = (-1)*λ(Vons(PART))*Q(PART) *2 *Vons(PART)*Rpcs(POIo(t),t)^(-2 - 1)*cos(Aθpc(POIo(t),t)) (mathH)/* for use in differentiations /% ∂[∂(t): K2] = (-2)*λ(Vons(PART))*Q(PART)*Vons(PART)*Rpcs(POIo(t),t)^(-3)*cos(Aθpc(POIo(t),t)) (endMath) /********************* >>>>>>>>> ∂[∂(t): K1] for use in differentiations From "K0, K1, K2, K3 for integro-differential equations" 03Oct2019 I need to check this before using it!! /% K1 = -3*β^2*Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(-5)*Vons(PART)*t*(cos(Aθpc(POIo(t),t=0)) - 1) ∂[∂(t): K1] = ∂[∂(t): 3*β^2 *Rocs(POIo)^2*Rpcs(POIo(t),t)^(-5)*Q(PART)*(-1)*Vons(PART)*t*(cos(Aθpc(POIo(t),t)) - 1) ] = -3*β^2*Vons(PART)*Rocs(POIo)^2*Q(PART)*∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1) ] /* insert derivative expressions from "Howell - Background math for Lucas Universal Force, Chapter 4.txt" section "∂[∂(t): Rpcs(POIo(t),t)^(-b)*sin(Aθpc(POIo(t),t))^a]" section "∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)]" /% (mathH)/* ???? where is the section on this? /% ∂[∂(t): K1] = -3*β^2*Vons(PART)*Rocs(POIo)^2*Q(PART) * 0 (endMath) /* Note the zeroing of the derivative term under several HIGHLY restrictive conditions!!! - RFp = RFo, particle is at origin (both reference frames) - for a POIo along direction of flight of particle, so cos(Aθpc(POIo(t),t)) = 1 - t=0 therefore : /% (mathH)/* when [t=0, RFp=RFo @t=0], maybe use only AFTER differentiations??? /% ∂[∂(t): K1(t=0)] = 0 (endMath) /********************* >>>>>>>>> ∂[∂(t): K0(t=0),K2(t=0)] in terms of E0ods(POIo,t) for relativistic factor, when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! /% ????:(mathH) ∂[∂(t): K0] = 21/2*β^2*Rocs(POIo)^3*Q(PART)*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) for use in differentiations ∂[∂(t): K0(t=0)] = 21/2*β^2*Rocs(POIo)^3*Q(PART)*Vons(PART)*sin(Aθpc(POIo(t),t=0))^2*cos(Aθpc(POIo(t),t=0))*Rpcs(POIo(t),t=0)^(-6) 1049:(mathH) Rocs(POIo) = Rpcs(POIo(t),t=0) when : t=0, RFp=RFo @t=0 = 21/2*β^2*Q(PART) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) 1118:(mathH) E0ods(POIo,t) = E0pds(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2 = E0ods(POIo,t) *21/2*β^2*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-1) (mathH)/* when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! /% ∂[∂(t): K0(t=0)] = E0ods(POIo,t)*21/2*β^2*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-1) (endMath) ????:(mathH) ∂[∂(t): K2] = (-2)*λ(Vons(PART))*Q(PART)*Vons(PART)*Rpcs(POIo(t),t)^(-3)*cos(Aθpc(POIo(t),t)) for use in differentiations ∂[∂(t): K2(t=0)] = (-2)*λ(Vons(PART))*Q(PART)*Vons(PART)*Rpcs(POIo(t),t=0)^(-3)*cos(Aθpc(POIo(t),t=0)) 1118:(mathH) E0ods(POIo,t) = E0pds(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2 = (-2)*λ(Vons(PART)) *Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIo(t),t=0)) (mathH)/* when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! /% ∂[∂(t): K2(t=0)] = (-2)*λ(Vons(PART))*Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIo(t),t=0)) (endMath) /********************* >>>>>>>>> ∂[∂(t): K_1st,K_2nd,K_3rd] for use in differentiations 1042:(mathH) K_1st = 3/2*β^2*Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t=0)^(-5)*sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART))*Q(PART)*Rpcs(POIo(t),t=0)^(-2) for use in differentiations ∂[∂(t): K_1st] = ∂[∂(t): 3/2*β^2*Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t=0)^(-5)*sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART))*Q(PART)*Rpcs(POIo(t),t=0)^(-2)] = + ∂[∂(t): 3/2*β^2*Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t=0)^(-5)*sin(Aθpc(POIo(t),t=0))^2] - ∂[∂(t): λ(Vons(PART))*Q(PART)*Rpcs(POIo(t),t=0)^(-2)] /* percolate constants /% = + 3/2*β^2*Q(PART)*Rocs(POIo)^3 *∂[∂(t): Rpcs(POIo(t),t=0)^(-5)*sin(Aθpc(POIo(t),t=0))^2] - λ(Vons(PART))*Q(PART) *∂[∂(t): Rpcs(POIo(t),t=0)^(-2)] 2110:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-5)] = 7*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) 2100:(mathH) ∂[∂(t): Rpcs(POIo(t),t)^(-2)] = 2*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) = + 3/2*β^2*Q(PART)*Rocs(POIo)^3 *7*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) - λ(Vons(PART))*Q(PART) *2*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) (mathH)/* for use in differentiations /% ∂[∂(t): K_1st] = 21/2*β^2*Q(PART)*Rocs(POIo)^3*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) - 2*λ(Vons(PART))*Q(PART)*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) (endMath) /* Check /% 1038:(mathH) K_1st = K0 + K2 for use in differentiations /* therefore /% ∂[∂(t): K_1st] = ∂[∂(t): K0 + K2] = ∂[∂(t): K0] + ∂[∂(t): K2] 2389:(mathH) ∂[∂(t): K0] = 21/2*β^2*Rocs(POIo)^3*Q(PART)*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) for use in differentiations 2398:(mathH) ∂[∂(t): K2] = (-2)*λ(Vons(PART))*Q(PART)*Vons(PART)*Rpcs(POIo(t),t)^(-3)*cos(Aθpc(POIo(t),t)) for use in differentiations = 21/2*β^2*Rocs(POIo)^3*Q(PART)*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) + (-2)*λ(Vons(PART))*Q(PART)*Vons(PART)*Rpcs(POIo(t),t)^(-3)*cos(Aθpc(POIo(t),t)) = 21/2*β^2*Rocs(POIo)^3*Q(PART)*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) - 2*λ(Vons(PART))*Q(PART)*Vons(PART)*Rpcs(POIo(t),t)^(-3)*cos(Aθpc(POIo(t),t)) = 21/2*β^2*Q(PART)*Rocs(POIo)^3*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) - 2*λ(Vons(PART))*Q(PART)*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) /* OK - this works /% 1063:(mathH) K_2nd = 21/8*β^4*Q(PART)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4 - λ(Vons(PART))*β^2*Q(PART)*Rocs(POIo)*Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t=0))^2 for use in differentiations ∂[∂(t): K_2nd] = ∂[∂(t): 21/8*β^4*Q(PART)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4 - λ(Vons(PART))*β^2*Q(PART)*Rocs(POIo)*Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t=0))^2] = + ∂[∂(t): 21/8*β^4*Q(PART)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4] - ∂[∂(t): λ(Vons(PART))*β^2*Q(PART)*Rocs(POIo)*Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t=0))^2] /* percolate constants /% = + 21/8*β^4*Q(PART) *∂[∂(t): Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4] - λ(Vons(PART))*β^2*Q(PART)*Rocs(POIo) *∂[∂(t): Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t=0))^2] 2114:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-6)] = 10*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) 2106:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-3)] = 5*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) = + 21/8*β^4*Q(PART) *10*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) - λ(Vons(PART)) *β^2*Q(PART)*Rocs(POIo) *5*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) = + 210/8*β^4*Q(PART) *Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) - λ(Vons(PART))*5*β^2*Q(PART)*Rocs(POIo) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) (mathH)/* for use in differentiations /% ∂[∂(t): K_2nd] = + 210/8*β^4*Q(PART) *Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) - λ(Vons(PART))*5*β^2*Q(PART)*Rocs(POIo) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) (endMath) /********************* >>>>>>>>> ∂[∂(t): K_1st(t=0),K_2nd(t=0),K_3rd(t=0)] in terms of E0ods(POIo,t) for relativistic factor, when [t=0, RFp=RFo @t=0], use only AFTER differentiations!!! /% 2468:(mathH) ∂[∂(t): K_1st] = 21/2*β^2*Q(PART)*Rocs(POIo)^3*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) - 2*λ(Vons(PART))*Q(PART)*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) for use in differentiations ∂[∂(t): K_1st(t=0)] = 21/2*β^2*Q(PART)*Rocs(POIo)^3*Vons(PART)*sin(Aθpc(POIo(t),t=0))^2*cos(Aθpc(POIo(t),t=0))*Rpcs(POIo(t),t=0)^(-6) - λ(Vons(PART))*2*Q(PART)*Vons(PART)*cos(Aθpc(POIo(t),t=0))*Rpcs(POIo(t),t=0)^(-3) 1049:(mathH) Rocs(POIo) = Rpcs(POIo(t),t=0) when : t=0, RFp=RFo @t=0 = 21/2*β^2 *Q(PART)*Vons(PART)*sin(Aθpc(POIo(t),t=0))^2*cos(Aθpc(POIo(t),t=0))*Rpcs(POIo(t),t=0)^(-3) - λ(Vons(PART))*2*Q(PART)*Vons(PART) *cos(Aθpc(POIo(t),t=0))*Rpcs(POIo(t),t=0)^(-3) 1109:(mathH) E0ods(POIo,t) = E0pds(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2 = E0ods(POIo,t) *21/2*β^2 *Vons(PART)*sin(Aθpc(POIo(t),t=0))^2*cos(Aθpc(POIo(t),t=0))*Rpcs(POIo(t),t=0)^(-1) - E0ods(POIo,t)*λ(Vons(PART)) *2 *Vons(PART) *cos(Aθpc(POIo(t),t=0))*Rpcs(POIo(t),t=0)^(-1) (mathH)/* when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! /% ∂[∂(t): K_1st(t=0)] = E0ods(POIo,t) *21/2*β^2*Vons(PART)*sin(Aθpc(POIo(t),t=0))^2*cos(Aθpc(POIo(t),t=0))*Rpcs(POIo(t),t=0)^(-1) - E0ods(POIo,t)*λ(Vons(PART)) *2 *Vons(PART) *cos(Aθpc(POIo(t),t=0))*Rpcs(POIo(t),t=0)^(-1) (endMath) 2497:(mathH) ∂[∂(t): K_2nd] = 210/8*β^4*Q(PART)*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) - λ(Vons(PART))*5*β^2*Q(PART)*Rocs(POIo)*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) for use in differentiations ∂[∂(t): K_2ndt=0)] >> 04Oct2019 wait for later - I might not need this? 1049:(mathH) Rocs(POIo) = Rpcs(POIo(t),t=0) when : t=0, RFp=RFo @t=0 1109:(mathH) E0ods(POIo,t) = E0pds(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2 /********************* >>>>>>>>> ∂[∂(t): E0pds(POIp)] using /% 1065:(mathH) E0pds(POIp) = Q(PART)/Rpcs(POIp)^2 /* Because there is no change in the static electric field in RFp coordinates : /% (mathH) ∂[∂(t): E0pds(POIp)] = 0 (endMath) /********************* >>>>>>>>> ∂[∂(t): E0ods(POIo,t)] In [Maxwell, relativity] electrodynamics, time delays are ignored!! (wrong!) Therefore, the STATIC E field is the same in RFp and RFo coordinates. /* using /% 1077:(mathH) E0ods(POIo,t) = E0pds(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2 ∂[∂(t): E0ods(POIo,t)] = ∂[∂(t): Q(PART)/Rpcs(POIo(t),t)^2] = Q(PART) *∂[∂(t): Rpcs(POIo(t),t)^(-2)] 1377:(mathH) ∂[∂(t): Rpcs(POIo(t),t)^(-α)] = α*Vons(PART)*Rpcs(POIo(t),t)^(-α - 1)*cos(Aθpc(POIo(t),t)) = Q(PART)*α*Vons(PART)*Rpcs(POIo(t),t)^(-α - 1)*cos(Aθpc(POIo(t),t)) = Q(PART)*2*Vons(PART)*Rpcs(POIo(t),t)^(-2 - 1)*cos(Aθpc(POIo(t),t)) (mathH) ∂[∂(t): E0ods(POIo,t)] = 2*Q(PART)*Vons(PART)*Rpcs(POIo(t),t)^(-3)*cos(Aθpc(POIo(t),t)) (endMath) /* putting into form with E0ods(POIo,t), again using /% 1077:(mathH) E0ods(POIo,t) = E0pds(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2 (mathH) ∂[∂(t): E0ods(POIo,t)] = E0ods(POIo,t) *2*Vons(PART)*Rpcs(POIo(t),t)^(-1)*cos(Aθpc(POIo(t),t)) (endMath) /********************* >>>>>>>>> Summary of ith stage EIods(POIo,t,ith stage)) derivations /*/*$ cat >>"$p_augmented" "$d_augment""d-dt EIods - summary of ith stages.txt" /* Note that f_sphereCapSurf(EIods(POIo,t=0)) is the round-off error for each stage, and the term containing it is dropped before progressing to the for symbols & HIGHLY restrictive conditions : see "textIn - relativistic factor, intermediate symbols, restrictive conditions.txt" Also note : /% K_1st = + 3/2*β^2*Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t=0)^(-5)*sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART)) *Q(PART) *Rpcs(POIo(t),t=0)^(-2) f_sphereCapSurf{x} = β*Rocs(POIo)^2*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))*∂[∂(t): x] ] /% >>>>>>>>>>>> generative form 2425:(mathL)/* generative form /% EIods(POIo,t,1st stage) = K_1st + f_sphereCapSurf(EIods(POIo,t,0th stage)) 2454:(mathL)/* generative form /% EIods(POIo,t,2nd stage) = K_1st + f_sphereCapSurf(EIods(POIo,t,1st stage)) 3081:(mathL)/* generative form /% EIods(POIo,t,3rd stage) = K_1st + f_sphereCapSurf(EIods(POIo,t,2nd stage)) 3368:(mathL)/* generative form /% EIods(POIo,t,4th stage) = K_1st + f_sphereCapSurf(EIods(POIo,t,3rd stage)) >>>>>>>>>>>> differentiable form 2441:(mathL)/* differentiable form /% EIods(POIo,t,1st stage) = + Q(PART) *( + 3/2 *β^2*Rocs(POIo)^3 *sin(Aθpc(POIo(t),t))^2 *Rpcs(POIo(t),t)^(-5) - λ(Vons(PART)) *1 ) + f_sphereCapSurf{EIods(POIo,t)} 3027:(mathL)/* differentiable form /% EIods(POIo,t,2nd stage) = + Q(PART) *( + 3/2 *β^2*Rocs(POIo)^3 *sin(Aθpc(POIo(t),t))^2 *Rpcs(POIo(t),t)^(-5) + 21/8 *β^4*Rocs(POIo)^4 *sin(Aθpc(POIo(t),t))^4 *Rpcs(POIo(t),t)^(-6) - λ(Vons(PART)) *1 *Rpcs(POIo(t),t)^(-2) - λ(Vons(PART)) *1 *β^2*Rocs(POIo) *sin(Aθpc(POIo(t),t))^2 *Rpcs(POIo(t),t)^(-3) ) + f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]} 3284:(mathL)/* differentiable form /% EIods(POIo,t,3rd stage) = + Q(PART) *( + 3/2 *β^2*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2 + 21/8 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^4 + 35/8 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(-7)*sin(Aθpc(POIo(t),t))^6 - λ(Vons(PART)) *1 *Rpcs(POIo(t),t)^(-2) - λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t))^2 - λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^4 ) + f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}} 3579:(mathL)/* differentiable form /% EIods(POIo,t,4th stage) = + Q(PART) *( + 3/2 *β^2*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2 + 21/8 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^4 + 35/8 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(-7)*sin(Aθpc(POIo(t),t))^6 + 455/64*β^8*Rocs(POIo)^6 *Rpcs(POIo(t),t)^(-8)*sin(Aθpc(POIo(t),t))^8 - λ(Vons(PART)) *1 *Rpcs(POIo(t),t)^(-2) - λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t))^2 - λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^4 - λ(Vons(PART)) *5/3 *β^6*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^6 ) + f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}}} >>>>>>>>>>>> HIGHLY restrictive conditions : t=0, RFo(t=0) = Rfp(t=0), Rocs(POIo) = Rpcs(POIo(t),t=0) 3016:(mathL)/* HIGHLY restrictive conditions : t=0, RFo(t=0) = Rfp(t=0), Rocs(POIo) = Rpcs(POIo(t),t=0) This means that the [observer, particle] reference frames are exactly the same at t=0 (other than motion). drop as roundoff error : f_sphereCapSurf expression see "textIn - relativistic factor, intermediate symbols, restrictive conditions.txt" /% EIods(POIo,t=0,2nd stage) = E0ods(POIo,t) *{ 3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 + 21/8*β^4*sin(Aθpc(POIo(t),t=0))^4} - E0ods(POIo,t)*λ(Vons(PART))*{1 + β^2*sin(Aθpc(POIo(t),t=0))^2} 3242:(mathL)/* HIGHLY restrictive conditions : t=0, RFo(t=0) = Rfp(t=0), Rocs(POIo) = Rpcs(POIo(t),t=0) This means that the [observer, particle] reference frames are exactly the same at t=0 (other than motion). drop as roundoff error : f_sphereCapSurf expression see "textIn - relativistic factor, intermediate symbols, restrictive conditions.txt" /% EIods(POIo,t=0,3rd stage) = + E0pds(POIp) *{ 3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 + 3/2*β^4*sin(Aθpc(POIo(t),t=0))^4 + 21/8 *β^6*sin(Aθpc(POIo(t),t=0))^6} - E0pds(POIp)*λ(Vons(PART)) *{1 + β^2*sin(Aθpc(POIo(t),t=0))^2 - β^4*sin(Aθpc(POIo(t),t=0))^4} 3639:(mathL)/* HIGHLY restrictive conditions : t=0, RFo(t=0) = Rfp(t=0), Rocs(POIo) = Rpcs(POIo(t),t=0) This means that the [observer, particle] reference frames are exactly the same at t=0 (other than motion). drop as roundoff error : f_sphereCapSurf expression see "textIn - relativistic factor, intermediate symbols, restrictive conditions.txt" /% EIods(POIo,t,4th stage) = + E0pds(POIp) *{ 3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 + 21/8*β^4*sin(Aθpc(POIo(t),t=0))^4 + 35/8*β^6*sin(Aθpc(POIo(t),t=0))^6 + 455/64*β^8*sin(Aθpc(POIo(t),t=0))^8} - E0pds(POIp)*λ(Vons(PART)) *{1 + 1* β^2*sin(Aθpc(POIo(t),t=0))^2 + 5/4 *β^4*sin(Aθpc(POIo(t),t=0))^4 + 5/3 *β^6*sin(Aθpc(POIo(t),t=0))^6} ????:(mathL)/* theoretical target - binomial series /% EIods(POIo,t,4th stage) = + E0pds(POIp) *{ 3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 + 3/8*β^4*sin(Aθpc(POIo(t),t=0))^4 + -1/16*β^6*sin(Aθpc(POIo(t),t=0))^6 + 3/128*β^8*sin(Aθpc(POIo(t),t=0))^8} - E0pds(POIp)*λ(Vons(PART)) *{1 + 3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 + 3/8*β^4*sin(Aθpc(POIo(t),t=0))^4 + -1/16*β^6*sin(Aθpc(POIo(t),t=0))^6} (endMath) ????:(mathL) binomial series = 1 3/2 3/8 -1/16 3/128 -3/256 7/1024 Lucas = 1 3/2 15/8 35/16 12Oct2019 Howell ETpds(POIp) = 1 3/2 21/8 35/8 455/64 ETpds(POIp)*λ(Vons(PART)) = 1 1 5/4 5/3 (endMath) /*_endCmd /********************* >>>>>>>>> Summary of ith stage ETods(POIo,t,ith stage)) The change from EIods(POIo,t,ith stage)) to ETods(POIo,t,ith stage)) is simple - just add a "one" to the (β*sin)^n series for E0pds(POIp) (1st expression RHS). Example for the 2nd stage is given below. /% 3016:(mathL)/* HIGHLY restrictive conditions : t=0, RFo(t=0) = Rfp(t=0), Rocs(POIo) = Rpcs(POIo(t),t=0) This means that the [observer, particle] reference frames are exactly the same at t=0 (other than motion). drop as roundoff error : f_sphereCapSurf expression see "textIn - relativistic factor, intermediate symbols, restrictive conditions.txt" /% EIods(POIo,t=0,2nd stage) = E0ods(POIo,t) *{ 3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 + 21/8*β^4*sin(Aθpc(POIo(t),t=0))^4} - E0ods(POIo,t)*λ(Vons(PART))*{1 + β^2*sin(Aθpc(POIo(t),t=0))^2} 3026:(mathL)/* HIGHLY restrictive conditions : t=0, RFo(t=0) = Rfp(t=0), Rocs(POIo) = Rpcs(POIo(t),t=0) This means that the [observer, particle] reference frames are exactly the same at t=0 (other than motion). drop as roundoff error : f_sphereCapSurf expression see "textIn - relativistic factor, intermediate symbols, restrictive conditions.txt" /% ETods(POIo,t=0,2nd stage) = E0ods(POIo,t) *{1 + 3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 + 21/8*β^4*sin(Aθpc(POIo(t),t=0))^4} - E0ods(POIo,t)*λ(Vons(PART))*{1 + β^2*sin(Aθpc(POIo(t),t=0))^2} /********************* >>>>>>>>> ∂[∂(t): K_1st + f_sphereCapSurf(EIods(POIo,t=0,(i-1) stage))) ] >>>>>>>>>>>> General process 1. derive ∂[∂(t): f_sphereCapSurf(EIods(POIo,t,(i-1) stage))) ] 2. express full K_1st + f_sphereCapSurf(EIods(POIo,t=0,(i-1) stage))) using result from 1 above and standard ∂[∂(t): K_1st] 2669:(mathH)/* for use in differentiations /% ∂[∂(t): K_1st] = 21/2*β^2*Q(PART)*Rocs(POIo)^3*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) - 2*λ(Vons(PART))*Q(PART)*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) 3. adjust to get integrable form by substiting for t=0 to get ∂[∂(t): K_1st + f_sphereCapSurf(EIods(POIo,t=0,(i-1) stage))) ] ********************* >>>>>>>>> ∂[∂(t): E0ods(POIo,t)*sin(Aθpc(POIo(t),t))^a] re-checked 03Oct2019 /*/*$ cat >>"$p_augmented" "$d_augment""d-dt E0ods*sin^a.txt" +-----+ Derivation : ∂[∂(t): E0ods(POIo,t) *sin(Aθpc(POIo(t),t))^a] = ∂[∂(t): E0ods(POIo,t)] *sin(Aθpc(POIo(t),t))^a + E0ods(POIo,t)*∂[∂(t): sin(Aθpc(POIo(t),t))^a] = ∂[∂(t): E0ods(POIo,t)] *sin(Aθpc(POIo(t),t))^a + E0ods(POIo,t) *∂[∂(t): sin(Aθpc(POIo(t),t))^a] /* using /% 2467:(mathH) ∂[∂(t): E0ods(POIo,t)] = E0ods(POIo,t) *2*Vons(PART)*Rpcs(POIo(t),t)^(-1)*cos(Aθpc(POIo(t),t)) ∂[∂(t): E0ods(POIo,t) *sin(Aθpc(POIo(t),t))^a] = E0ods(POIo,t)*2*Vons(PART)*Rpcs(POIo(t),t)^(-1)*cos(Aθpc(POIo(t),t)) *sin(Aθpc(POIo(t),t))^a + E0ods(POIo,t) *a*sin(Aθpc(POIo(t),t))^(a-1)*cos(Aθpc(POIo(t),t)) /* rearranging /% ∂[∂(t): E0ods(POIo,t) *sin(Aθpc(POIo(t),t))^a] = E0ods(POIo,t) *2*Vons(PART)*Rpcs(POIo(t),t)^(-1) *cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^a + E0ods(POIo,t) *a *cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a-1) /* Normally, in 4-[32 to 37] t=0 applies to resultant expressions, and restrictive conditions apply to this result! : (mathH) ∂[∂(t): E0ods(POIo,t)*sin(Aθpc(POIo(t),t))^a] = E0ods(POIo,t) *2*Vons(PART)*Rpcs(POIo(t),t)^(-1) *cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^a + E0ods(POIo,t) *a *cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a-1) (endMath) +-----+ /*Specific results (see a few derivations further below) : see "Binomial Series for Chapter 4.ods" /% (mathH) ∂[∂(t): E0ods(POIo,t=0)*sin(Aθpc(POIo(t),t=0))^1] = E0ods(POIo,t=0)*2*Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIp(t),t=0))*sin(Aθpc(POIp(t),t=0))^1 + E0ods(POIo,t=0)*1*cos(Aθpc(POIo(t),t=0))*sin(Aθpc(POIo(t),t=0))^0 (endMath) (mathH) ∂[∂(t): E0ods(POIo,t=0)*sin(Aθpc(POIo(t),t=0))^2] = E0ods(POIo,t=0)*2*Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIp(t),t=0))*sin(Aθpc(POIp(t),t=0))^2 + E0ods(POIo,t=0)*2*cos(Aθpc(POIo(t),t=0))*sin(Aθpc(POIo(t),t=0))^1 (endMath) (mathH) ∂[∂(t): E0ods(POIo,t=0)*sin(Aθpc(POIo(t),t=0))^3] = E0ods(POIo,t=0)*2*Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIp(t),t=0))*sin(Aθpc(POIp(t),t=0))^3 + E0ods(POIo,t=0)*3*cos(Aθpc(POIo(t),t=0))*sin(Aθpc(POIo(t),t=0))^2 (endMath) (mathH) ∂[∂(t): E0ods(POIo,t=0)*sin(Aθpc(POIo(t),t=0))^4] = E0ods(POIo,t=0)*2*Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIp(t),t=0))*sin(Aθpc(POIp(t),t=0))^4 + E0ods(POIo,t=0)*4*cos(Aθpc(POIo(t),t=0))*sin(Aθpc(POIo(t),t=0))^3 (endMath) (mathH) ∂[∂(t): E0ods(POIo,t=0)*sin(Aθpc(POIo(t),t=0))^5] = E0ods(POIo,t=0)*2*Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIp(t),t=0))*sin(Aθpc(POIp(t),t=0))^5 + E0ods(POIo,t=0)*5*cos(Aθpc(POIo(t),t=0))*sin(Aθpc(POIo(t),t=0))^4 (endMath) (mathH) ∂[∂(t): E0ods(POIo,t=0)*sin(Aθpc(POIo(t),t=0))^6] = E0ods(POIo,t=0)*2*Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIp(t),t=0))*sin(Aθpc(POIp(t),t=0))^6 + E0ods(POIo,t=0)*6*cos(Aθpc(POIo(t),t=0))*sin(Aθpc(POIo(t),t=0))^5 (endMath) (mathH) ∂[∂(t): E0ods(POIo,t=0)*sin(Aθpc(POIo(t),t=0))^8] = E0ods(POIo,t=0)*2*Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIp(t),t=0))*sin(Aθpc(POIp(t),t=0))^8 + E0ods(POIo,t=0)*8*cos(Aθpc(POIo(t),t=0))*sin(Aθpc(POIo(t),t=0))^7 (endMath) /*_endCmd /********************* >>>>>>>>> ∂[∂(t): E0ods(POIo,t=0)*Rpcs(POIo(t),t)^(-β)*sin(Aθoc(POIo))^a] 03Oct2019 - this is very old and must be [checked, modified] if it is to be used!! get rid of all "Aθtc" -> Aθ[o,p]c 08Jun2016 - Can I prove that the term with Vons << the second term, so that it may be dropped? (I doubt it very much!! except for very special conditions!) - 1/c^n terms does it, but makes Lucas's work irrelevant. - I must have screwed up again somewhere... /% /*/*$ cat >>"$p_augmented" "$d_augment""d-dt E0ods*Rpcs^-β*sin^a.txt" /* Reference frame notations for Chapter 4 (constant Vonv(PART)) - most commonly used forms in (4-30) to (4-37) : /* constants : c,Q,Vons,beta,lambda(v)=beta^2 particle RFp : t , Rocs(POIp(t),t),Rpcs(POIp) ,Aθoc(POIp(t),t),Aθpc(POIp),EIpds(POIp) ,[EI,BX] =0 in RFp observer RFo : t , Rocs(POIo) ,Rpcs(POIo(t),t) ,Aθoc(POIo) ,Aθpc(POIo(t),t),EIods(POIo) ,BIods(POIo) frozen RFt : t=0, Rocs(POIo) ,Rpcs(POIo(t),t), t=0),Aθtc(RFt) ,EItds(notused),BItds(notused) /* POIo context : integration ∫[∂(Aθpc),0.to.Aθpc(POIo(t), t=0): expressions with [Rocs(POIo),Rpcs(POIo(t), t=0),Aθtc(RFt)]} result gives [Rocs(POIo),Rpcs(POIo(t),t),Aθoc(POIo),Aθpc(POIo(t),t)] } /* For integrals in next step : derivative ∂[∂(t): expressions with [Rocs(POIo),Rpcs(POIo(t),t) ,Aθoc(POIo),Aθpc(POIo(t),t),E0ods(POIo,t)] result gives [Rocs(POIo),Rpcs(POIo(t),t=0),Aθtc(RFo) ,Aθtc(RFt) ,E0ods(POIo,t=0)] /* General formulae for Chapter 4 /$ dp[dt: E0ods(POIo,t=0)*Rpcs(POIo(t),t)^( - b)*sin(Aθoc(POIo))^a] = dp[dt: E0ods(POIo,t=0)] *Rpcs(POIo(t),t)^(-b)*sin(Aθoc(POIo))^a + E0ods(POIo,t=0) *dp[dt: Rpcs(POIo(t),t)^( - b)*sin(Aθoc(POIo))^a] /*From "Derivatives & Integrals adapted to Chapter 4 - Summary" above /% ∂[∂(t): Rpcs(POIo(t),t)^( - β)*sin(Aθpc(POIo(t),t))^a] = β*Vons(PART)*Rpcs(POIo(t),t)^(-β-1)*cos(Aθtc(POIp(t),t=0))*sin(Aθtc(RFt))^(a ) + a *Rpcs(POIo(t),t)^(-β )*cos(Aθtc(POIp(t),t=0))*sin(Aθtc(RFt))^(a-1) /* From "Howell - Background math for Lucas Universal Force, Chapter 4.odt" /% section "∂[∂(t): E0ods(POIo,t)] = ∂[∂(t): E0pds(POIo(t),t)] - proper E0odv(POIo,t) vector approach" 13*) ∂[∂(t): E0ods(POIo,t)] = 2*|Q(PART)|*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 = E0ods(POIo,t)*2*Vons(PART)*Rpcs(POIo(t),t)^(-1)*cos(Aθpc(POIo(t),t)) /* Put into RFt form to be fed into integral /* ∂[∂(t): E0ods(POIo,t)] = E0ods(POIo(t)=0)*2*Vons(PART)*Rpcs(POIo(t),t)^(-1)*cos(Aθpc(POIo(t),t)) /*Click to see Therefore : /% ∂[∂(t): E0ods(POIo,t=0)*Rpcs(POIo(t),t)^( - β)*sin(Aθoc(POIo))^a] = E0ods(POIo,t=0)* (+ 2 *Vons(PART)*Rpcs(POIo(t),t)^(-1 )*cos(Aθtc(POIp(t),t=0)) *Rpcs(POIo(t),t)^(-β)*sin(Aθtc(RFt))^a + β *Vons(PART)*Rpcs(POIo(t),t)^(-β-1)*cos(Aθtc(POIp(t),t=0))*sin(Aθtc(RFt))^(a ) + a *Rpcs(POIo(t),t)^(-β )*cos(Aθtc(POIp(t),t=0))*sin(Aθtc(RFt))^(a-1) () = E0ods(POIo,t=0)* (+ 2 *Vons(PART)*Rpcs(POIo(t),t)^(-1 )*cos(Aθtc(POIp(t),t=0)) *Rpcs(POIo(t),t)^(-β)*sin(Aθtc(RFt))^a + β *Vons(PART)*Rpcs(POIo(t),t)^(-β-1)*cos(Aθtc(POIp(t),t=0))*sin(Aθtc(RFt))^(a ) + a *Rpcs(POIo(t),t)^(-β )*cos(Aθtc(POIp(t),t=0))*sin(Aθtc(RFt))^(a-1) () /* General formulae for Chapter 4 ∂[∂(t): E0ods(POIo(t)=0) *Rpcs(POIo(t),t)^(-b)*sin(Aθoc(POIo))^a] = E0ods(POIo(t)=0)* (+ 2 *Vons(PART)*Rpcs(POIo(t),t)^(-1-b)*cos(AOtc(RFt))*sin(AOtc(RFt))^a + b *Vons(PART)*Rpcs(POIo(t),t)^(-b-1)*cos(AOtc(RFt))*sin(AOtc(RFt))^a + a *Rpcs(POIo(t),t)^(-b )*cos(AOtc(RFt))*sin(AOtc(RFt))^(a-1) () = (+ E0ods(POIo(t)=0)*(2+b)*Vons(PART)*Rpcs(POIo(t)=0)^(-1-b)*cos(AOtc(RFt))*sin(AOtc(RFt))^a + E0ods(POIo(t)=0)*a *Rpcs(POIo(t)=0)^(-b )*cos(AOtc(RFt))*sin(AOtc(RFt))^(a-1) (mathH)/* old-wrong!! /% ∂[∂(t): E0ods(POIo,t=0)*Rpcs(POIo(t),t)^(-β)*sin(Aθoc(POIo))^a] = + E0ods(POIo(t)=0)*(2+b)*Vons(PART) *Rpcs(POIo(t)=0)^(-1-b)*cos(AOtc(RFt))*sin(AOtc(RFt))^a + E0ods(POIo(t)=0)*a *Rpcs(POIo(t)=0)^(-b )*cos(AOtc(RFt))*sin(AOtc(RFt))^(a-1) (endMath) /*-----+ /* Looking at "∂[∂(t): E0ods(POIo(t)=0)*Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4]" ∂[∂(t): E0ods(POIo(t)=0) *Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4] = ∂[∂(t): E0ods(POIo(t)=0)] *Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4 + E0ods(POIo(t)=0)*∂[∂(t): Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4] From "Derivatives & Integrals adapted to Chapter 4 - Summary" above ∂[∂(t): Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4 ] = 1*Vons(PART)*Rpcs(POIo(t),t)^(-2)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^4 + 4 *Rpcs(POIo(t),t)^(-1)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3 /* using /% 2462:(mathH) ∂[∂(t): E0ods(POIo,t)] = 2*Q(PART)*Vons(PART)*Rpcs(POIo(t),t)^(-3)*cos(Aθpc(POIo(t),t)) ∂[∂(t): E0ods(POIo(t)=0) *Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4] = ∂[∂(t): E0ods(POIo(t)=0)] *Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4 + E0ods(POIo(t)=0) *∂[∂(t): Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4] = 2*|Q(PART)|*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 *Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4 + E0ods(POIo(t)=0) *[ 1*Vons(PART)*Rpcs(POIo(t),t)^(-2)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^4 + 4 *Rpcs(POIo(t),t)^(-1)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3 ] (1) = 2*|Q(PART)|*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 *Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4 + E0ods(POIo(t)=0) *[ 1*Vons(PART)*Rpcs(POIo(t),t)^(-2)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^4 + 4 *Rpcs(POIo(t),t)^(-1)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3 ] /* From "E0ods(POIo,t) = E0pds(POIo(t),t)" 2*) E0ods(POIo,t) = |Q(PART)|/Rpcs(POIp(t),t)^2 /* Subbing (2*) into (1) and consolidating ∂[∂(t): E0ods(POIo(t)=0) *Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4] = E0ods(POIo,t)* (+ 2*Vons(PART)*Rpcs(POIo(t),t)^(-2)*cos(Aθpc(POIo(t),t))*sin(Aθoc(POIo))^4 + 1*Vons(PART)*Rpcs(POIo(t),t)^(-2)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^4 + 4 *Rpcs(POIo(t),t)^(-1)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3 ) = E0ods(POIo,t)* (+ 3*Vons(PART)*Rpcs(POIo(t),t)^(-2)*cos(Aθpc(POIo(t),t))*sin(Aθoc(POIo))^4 + 4 *Rpcs(POIo(t),t)^(-1)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3 ) /*----+ Check on general model : ∂[∂(t): E0ods(POIo(t)=0)*Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4] = E0ods(POIo(t)=0)* (+ 2 *Vons(PART)*Rpcs(POIo(t),t)^(-1-1)*cos(AOtc(RFt))*sin(AOtc(RFt))^4 + 1 *Vons(PART)*Rpcs(POIo(t),t)^(-1-1)*cos(AOtc(RFt))*sin(AOtc(RFt))^4 + 4 *Rpcs(POIo(t),t)^(-1 )*cos(AOtc(RFt))*sin(AOtc(RFt))^(4-1) () = E0ods(POIo(t)=0)* (+ 2 *Vons(PART)*Rpcs(POIo(t),t)^(-2)*cos(AOtc(RFt))*sin(AOtc(RFt))^4 + 1 *Vons(PART)*Rpcs(POIo(t),t)^(-2)*cos(AOtc(RFt))*sin(AOtc(RFt))^4 + 4 *Rpcs(POIo(t),t)^(-1)*cos(AOtc(RFt))*sin(AOtc(RFt))^3 () = E0ods(POIo(t)=0)* (+ 3 *Vons(PART)*Rpcs(POIo(t),t)^(-2)*cos(AOtc(RFt))*sin(AOtc(RFt))^4 + 4 *Rpcs(POIo(t),t)^(-1)*cos(AOtc(RFt))*sin(AOtc(RFt))^3 () OK!! This is the same as the earlier hand derivation : ∂[∂(t): E0ods(POIo(t)=0)*Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4] = E0ods(POIo,t)* (+ 3*Vons(PART)*Rpcs(POIo(t)=0)^(-2)*cos(AOtc(RFt))*sin(AOtc(RFt))^4 + 4 *Rpcs(POIo(t)=0)^(-1)*cos(AOtc(RFt))*sin(AOtc(RFt))^3 () /*_endCmd /********************* >>>>>>>>> ∂[∂(t): E0pdv(POIp) ] = ∂[∂(t): E0pdv(POIo(t),t) ] = 0 ≠ ∂[∂(t): E0odv(POIo)] From "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt", section ""Points of Interest" (POI)s", normally, E0pdv(POIp) and E0pdv(POIo(t),t) are NOT interchangeable, as the latter refers to a trajectory of a POIo over time as seen in the observer reference frame RFo. However, it is shown here in the context that within the particle frame of reference, E0, and therefore [BI, BT, EI, ET] do not change. As the static electric field (electrostatic field) in RFp is constant : /%(1) ∂[∂(t): E0pdv(POIo(t),t)] = 0 /********************* >>>>>>>>> ∂[∂(t): BIpdv(POIp) ] = ∂[∂(t): BIpdv(POIo(t),t) ] = ∂[∂(t): BTpdv(POIo(t),t) ] = 0 ≠ ∂[∂(t): BTodv(POIo,t)] From "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt", section ""Points of Interest" (POI)s", normally, BIpdv(POIp) and E0pdv(POIo(t),t) are NOT interchangeable, as the latter refers to a trajectory of a POIo over time as seen in the observer reference frame RFo. However, it is shown here in the context that within the particle frame of reference RFp, E0, and therefore [BI, BT, EI, ET] do not change, so BIpdf(POIp) and BTpdf(POIp) are NOT functions of time, and are equal. For the Chapter 4 situation, there is no "external" magnetic field (independent of the particle), and so B0 is a constant zero. As per the previous sub-section, the induced magnetic field is always zero, which means that its derivative is also zero in RFp. /%(1) ∂[∂(t): BIpdv(POIo(t),t)] = ∂[∂(t): BTpdv(POIo(t),t)] = 0 /*This is a strange comment - not one that I normally think of... endsection /********************************************** >>>>>> Integrals /********************* >>>>>>>>> ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^z] /% /*/*$ cat >>"$p_augmented" "$d_augment""∫_d_Aθpc, cos*sin^z.txt" ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^z] = sin(Aθpc(POIo(t),t=0))^(z+1)/(z+1) The following list makes it easier to copy/paste... Note that the result = 0 for the lower limit of integration, leaving only the upper end. /% (mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))] = sin(Aθpc(POIo(t),t=0))^1/1 (endMath) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))] = - {cos(Aθpc(POIo(t),t=0)) - cos(0)} =(-1)*{cos(Aθpc(POIo(t),t=0)) - 1} (mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))] = {1 - cos(Aθpc(POIo(t),t=0)} (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^1] = sin(Aθpc(POIo(t),t=0))^2/2 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^2] = sin(Aθpc(POIo(t),t=0))^3/3 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^3] = sin(Aθpc(POIo(t),t=0))^4/4 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^4] = sin(Aθpc(POIo(t),t=0))^5/5 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^5] = sin(Aθpc(POIo(t),t=0))^6/6 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^6] = sin(Aθpc(POIo(t),t=0))^7/7 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^7] = sin(Aθpc(POIo(t),t=0))^8/8 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^8] = sin(Aθpc(POIo(t),t=0))^9/9 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^9] = sin(Aθpc(POIo(t),t=0))^10/10 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^10] = sin(Aθpc(POIo(t),t=0))^11/11 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^11] = sin(Aθpc(POIo(t),t=0))^12/12 (endMath) /*_endCmd /********************* >>>>>>>>> ∫[∂(Aθpc),0.to.Aθpcf: Rpcs(POIo(t),t)^(-β)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a)] 30Aug2019 This section is probably irrelevant in light of the conclusions of : /* 30Aug2019 see "Howell - Background math for Lucas Universal Force, Chapter 4.txt" section '"Rpcs(POIo(t),t)" is a constant for integrals like "∫{∂(Aθpc),0.to.Aθoc(POIp(t),t=0):" ?' Rpcs(POIo(t),t=0) is a constant with respect ot that type of integral. /*/*$ cat >>"$p_augmented" "$d_augment""∫d_Aθpc, Rpcs^-β*cos*sin^a.txt" /*Approximate solution approach, here a,b >=0, element of integers : /%For integrals from Aθpc=0 to final, at Aθpc=0, sin(Aθpc(POIo(t),t))=0 and the expression is 0, producing a zero lower result for definite integrals. This was an attempt to [check, improve] results by look at Rpcs(POIo(t),t) as a variable of integration "with respect to" (wrt) Aθpc. However, that is not correct when following the circumference. /* General result from (5) below : /% ∫[∂(Aθpc),0.to.Aθpcf: Rpcs(POIo(t),t)^(-β)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^a} = 1/(a+β)/Rocs(POIo)*Rpcs(POIo(t),t)^(-β+1)*sin(Aθpc(POIo(t),t))^(a+1) /* List of special integrals /* 23Jun2016 not used in current versions as Rpcs(POIo(t)=0) is a "pseudo-constant" for integration wrt dAθpc 29Aug2019 - can switch from [∂(Aθpc),0.to.Aθpcf:, cos(Aθpc(POIo(t),t))] to [∂(Aθoc),0.to.Aθocf:, cos(Aθoc(POIp(t),t))] etc... /% (mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(-4)] = 1/ 4/Rocs(POIo)*Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t))^1 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(-5)] = 1/ 5/Rocs(POIo)*Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^1 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^1*Rpcs(POIo(t),t)^(-3)] = 1/ 4/Rocs(POIo)*Rpcs(POIo(t),t)^(-2)*sin(Aθpc(POIo(t),t))^2 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^1*Rpcs(POIo(t),t)^(-5)] = 1/ 6/Rocs(POIo)*Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^2 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-4)] = 1/ 6/Rocs(POIo)*Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t))^3 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-5)] = 1/ 7/Rocs(POIo)*Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^3 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3*Rpcs(POIo(t),t)^(-4)] = 1/ 7/Rocs(POIo)*Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t))^4 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3*Rpcs(POIo(t),t)^(-6)] = 1/ 9/Rocs(POIo)*Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^4 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3*Rpcs(POIo(t),t)^(-7)] = 1/10/Rocs(POIo)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^4 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-5)] = 1/ 9/Rocs(POIo)*Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^5 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-6)] = 1/10/Rocs(POIo)*Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^5 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^5*Rpcs(POIo(t),t)^(-5)] = 1/10/Rocs(POIo)*Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^6 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^5*Rpcs(POIo(t),t)^(-6)] = 1/11/Rocs(POIo)*Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^6 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^5*Rpcs(POIo(t),t)^(-7)] = 1/12/Rocs(POIo)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^6 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^6*Rpcs(POIo(t),t)^(-6)] = 1/12/Rocs(POIo)*Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^7 (endMath) (mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^7*Rpcs(POIo(t),t)^(-7)] = 1/14/Rocs(POIo)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^8 (endMath) /*-- First attempt : /% ∂[∂(Aθpc): sin(Aθpc(POIo(t),t))^(a + 1)*Rpcs(POIo(t),t)^( - β + 1)] = ∂[∂(Aθpc): sin(Aθpc(POIo(t),t))^(a + 1)] *Rpcs(POIo(t),t)^(-β+1) + sin(Aθpc(POIo(t),t))^(a+1) *∂[∂(Aθpc): Rpcs(POIo(t),t)^( - β + 1)] (1) = (a+1)*sin(Aθpc(POIo(t),t))^(a )*∂[∂(Aθpc): sin(Aθpc(POIo(t),t))] *Rpcs(POIo(t),t)^(-β+1) + sin(Aθpc(POIo(t),t))^(a+1) *(-β+1)*Rpcs(POIo(t),t)^(-β)*∂[∂(Aθpc): Rpcs(POIo(t),t)] From "∂[∂(Aθpc): Rpcs(POIo(t),t)] = ∂[∂(Aθpc): |Rpcv(POIo(t),t)|]" 2*) ∂[∂(Aθpc): Rpcs(POIo(t),t)] = (-1)*Rocs(POIo)*sin(Aθoc(POIo))*sin(Aθpc(POIo(t),t))^(-2)*cos(Aθpc(POIo(t),t)) Subbing (2*) into (1) : = ( a+1)*sin(Aθpc(POIo(t),t))^(a )*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-β+1) + (-β+1)*sin(Aθpc(POIo(t),t))^(a+1)*Rpcs(POIo(t),t)^(-β)*(-1)*Rocs(POIo)*sin(Aθoc(POIo))*sin(Aθpc(POIo(t),t))^(-2)*cos(Aθpc(POIo(t),t)) = ( a+1) *Rpcs(POIo(t),t)^(-β+1)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a ) + (-β+1)*(-1)*Rocs(POIo)*sin(Aθoc(POIo))*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a+1+1-2) = ( a+1) *Rpcs(POIo(t),t)^(-β ) *cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a ) *Rpcs(POIo(t),t)^(-β+1) + ( β-1)*Rocs(POIo)*sin(Aθoc(POIo)) *cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a-1) *Rpcs(POIo(t),t)^(-β ) (2) = cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a)*Rpcs(POIo(t),t)^(-β) * [ ( a+1)*Rpcs(POIo(t),t) + ( β-1)*Rocs(POIo)*sin(Aθoc(POIo))*sin(Aθpc(POIo(t),t))^(-1) } From "Rocv(POIo), Aθoc(POIo), Aφoc(POIo) are constants" : but thisd is with respect to time!! With respect to Aθpc(POIo(t),t), these are NOT constants (i.e POIo changes along trajectory of particle?) 5*) Rpcs (POIo(t),t)*sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) At t=0, the observer & particle reference frames coincide, and 3a) sin(Aθoc(POIo))|t=0 = sin(Aθpc(POIo(t),t))^(-1)|t=0 3b) Rocs(POIo) |t=0 = Rpcs(POIo(t),t) Subbing (5*),(3a),(3b) into (2) : (4) ∂[∂(Aθpc): sin(Aθpc(POIo(t),t))^(a + 1)*Rpcs(POIo(t),t)^( - β + 1)] = cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a) *Rpcs(POIo(t),t)^(-β) *[ (a+1)*Rpcs(POIo(t),t) + (β-1)*Rocs(POIo) } = cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a) *Rpcs(POIo(t),t)^(-β) *[ (a+1) + (β-1) ]*Rpcs(POIo(t),t) = cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a) *Rpcs(POIo(t),t)^(-β) * (a+β)*Rpcs(POIo(t),t) = (a+β)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a) *Rpcs(POIo(t),t)^(-β+1) For t=0, a,β >=0, ∈ integers. Also, at Aθpc=0, the expression is 0. As the upper endpoint of the integration at time t=0, Aθpc(POIo(t),t)), Rocs(POIo) Therefore : (5) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a)*Rpcs(POIo(t),t)^( - β)} = 1/(a+β)*sin(Aθpc(POIo(t),t))^(a+1)*Rpcs(POIo(t),t)^(-β+1) For t=0 when RFp & RFo coincide, a,β >=0, integers. POIo does NOT have to be on the perpendicular running through the origins. For integrals from Aθpc=0 to final, at Aθpc=0, sin(Aθpc(POIo(t),t))=0 and the expression is 0, producing a zero lower result for definite integrals. At Aθpc(POIo(t),t)) = 0, the expression is zero so the lower limit of integraion drops out. HOWEVER, at the upper limit of integration, @Aθpc(POIo(t),t)), Rpcs(POIo(t),t) = Rocs(POIo), so we cancel out their powers, result as Rpcs(POIo(t),t) (6) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a)*Rpcs(POIo(t),t)^( - β)} = 1/(a+β)*sin(Aθpc(POIo(t),t))^(a+1)*Rpcs(POIo(t),t)^(-β) For t=0 when RFp & RFo coincide, a,β >=0, integers. POIo does NOT have to be on the perpendicular running through the origins. For integrals from Aθpc=0 to final, at Aθpc=0, sin(Aθpc(POIo(t),t))=0 and the expression is 0, producing a zero lower result for definite integrals. At Aθpc(POIo(t),t)) = 0, the expression is zero so the lower limit of integraion drops out. /*-----+ Check on : /%∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(3)*Rpcs(POIo(t),t)^( - 7)} = sin(Aθpc(POIo(t),t))^(a+1)*Rpcs(POIo(t),t)^(-β+1)/(a+β)/Rocs(POIo) where a=3, β=7 = sin(Aθpc(POIo(t),t))^(3+1)*Rpcs(POIo(t),t)^(-7+1)/(a+β)/Rocs(POIo) (1) = sin(Aθpc(POIo(t),t))^(4) *Rpcs(POIo(t),t)^(-6) /(10 )/Rocs(POIo) Taking the derivative wrt Aθpc of (1) : (2) ∂[∂(Aθpc): sin(Aθpc(POIo(t),t))^(4)*Rpcs(POIo(t),t)^( - 6)/10/Rocs(POIo)] = [ 1/10/Rocs(POIo) }*∂[∂(Aθpc): sin(Aθpc(POIo(t),t))^(4)*Rpcs(POIo(t),t)^( - 6)] /*--+ Looking at the derivative dp[dAθpc : sin(Aθpc(POIo(t),t))^(4) *Rpcs(POIo(t),t)^(-6) ] = dp[dAθpc : sin(Aθpc(POIo(t),t))^(4) ] *Rpcs(POIo(t),t)^(-6) + sin(Aθpc(POIo(t),t))^(4)*dp[dAθpc : Rpcs(POIo(t),t)^(-6) ] = 4 *sin(Aθpc(POIo(t),t))^(3)*dp[dAθpc : sin(Aθpc(POIo(t),t)) ] *Rpcs(POIo(t),t)^(-6) + sin(Aθpc(POIo(t),t))^(4)*(-6)*Rpcs(POIo(t),t)^(-7)*dp[dAθpc : Rpcs(POIo(t),t) ] (3) = 4 *sin(Aθpc(POIo(t),t))^(3)*cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(-6) + (-6)*sin(Aθpc(POIo(t),t))^(4) *Rpcs(POIo(t),t)^(-7)*dp[dAθpc : Rpcs(POIo(t),t) ] /* From "dp[dAθpc : Rpcs(POIo(t),t) ] = dp[dAθpc : |Rpcv(POIo(t),t)|]" 2*) dp[dAθpc : Rpcs(POIo(t),t)] = (-1)*Rocs(POIo)*sin(Aθoc(POIo))*sin(Aθpc(POIo(t),t))^(-2)*cos(Aθpc(POIo(t),t)) Subbing into (3) : = 4 *cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(3)*Rpcs(POIo(t),t)^(-6) + (-6) *sin(Aθpc(POIo(t),t))^(4)*Rpcs(POIo(t),t)^(-7)*(-1)*Rocs(POIo)*sin(Aθoc(POIo))*sin(Aθpc(POIo(t),t))^(-2)*cos(Aθpc(POIo(t),t)) = 4 *cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(3)*Rpcs(POIo(t),t)^(-6) + 6 *cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(2)*Rpcs(POIo(t),t)^(-7) *Rocs(POIo)*sin(Aθoc(POIo)) (4) = cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(3)*Rpcs(POIo(t),t)^(-7) *[ 4*Rpcs(POIo(t),t) + 6*sin(Aθpc(POIo(t),t))^(-1) *Rocs(POIo)*sin(Aθoc(POIo)) } As in the previous sub-sub-section : From "Rocv(POIo), Aθoc(POIo), Aφoc(POIo) are constants" : 5*) Rpcs (POIo(t),t)*sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) At t=0, the observer & particle reference frames coincide, and 3a) sin(Aθoc(POIo))|t=0 = sin(Aθpc(POIo(t),t))^(-1)|t=0 3b) Rocs(POIo) |t=0 = Rpcs(POIo(t),t) Subbing into (4) : dp[dAθpc : sin(Aθpc(POIo(t),t))^(4)*Rpcs(POIo(t),t)^(-6) ] = cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(3)*Rpcs(POIo(t),t)^(-7)*Rocs(POIo)*( 4 + 6 ) = cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(3)*Rpcs(POIo(t),t)^(-7)*10*Rocs(POIo) Integrating both sides : ∫[dAθpc, 0 to Aθpcf : cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(3)*Rpcs(POIo(t),t)^(-7)*10*Rocs(POIo) } = sin(Aθpc(POIo(t),t))^(4)*Rpcs(POIo(t),t)^(-6) Removing constants from within the integral : ∫[dAθpc, 0 to Aθpcf : cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(3)*Rpcs(POIo(t),t)^(-7) } = sin(Aθpc(POIo(t),t))^(4)*Rpcs(POIo(t),t)^(-6)/10/Rocs(POIo) This is the same as the starting point at the start of this example : ∫[dAθpc, 0 to Aθpcf : cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(3)*Rpcs(POIo(t),t)^(-7) } = sin(Aθpc(POIo(t),t))^(4)*Rpcs(POIo(t),t)^(-6)/10/Rocs(POIo) confiming the general result for this example. /*-----+ Try this on expression missing a term : This example VIOLATES a condition for the general solution in that the sin(Aθpc(POIo(t),t))^(a) term is missing (a=0, when it should be >0 integer). However, I'm curious to try it out anyways. /* /% ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a)*Rpcs(POIo(t),t)^( - β)} = sin(Aθpc(POIo(t),t))^(a+1)*Rpcs(POIo(t),t)^(-β+1)/(a+β)/Rocs(POIo) /*For t=0 when RFp & RFo coincide, a,b >=0, integers. POIo does NOT have to be on the perpendicular running through the origins. For integrals from Aθpc=0 to final, at Aθpc=0, sin(Aθpc(POIo(t),t))=0 and the expression is 0, producing a zero lower result for definite integrals. /%At Aθpc(POIo(t),t)) = 0 & π/2, the expression is zero. /* For this example : /% ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^( - 5)} = sin(Aθpc(POIo(t),t))^(a+1)*Rpcs(POIo(t),t)^(-β+1)/(a+β)/Rocs(POIo) where a=0, β=5 (1) = sin(Aθpc(POIo(t),t))^(1)*Rpcs(POIo(t),t)^(-4)/5/Rocs(POIo) /*--+ Taking the derivative of the solution : dp[dAθpc : sin(Aθpc(POIo(t),t))^(1) *Rpcs(POIo(t),t)^(-4)/5/Rocs(POIo) ] = 1/5/Rocs(POIo)* { dp[dAθpc : sin(Aθpc(POIo(t),t))^(1)] *Rpcs(POIo(t),t)^(-4) ] + sin(Aθpc(POIo(t),t))^(1) *dp[dAθpc : Rpcs(POIo(t),t)^(-4) ] } (2) = 1/5/Rocs(POIo)* { (1) *sin(Aθpc(POIo(t),t))^(0)*dp[dAθpc : sin(Aθpc(POIo(t),t))] *Rpcs(POIo(t),t)^(-4) + sin(Aθpc(POIo(t),t))^(1) *(-4)*Rpcs(POIo(t),t)^(-5)*dp[dAθpc : Rpcs(POIo(t),t) ] } /* From "dp[dAθpc : Rpcs(POIo(t),t) ] = dp[dAθpc : |Rpcv(POIo(t),t)|]" 2*) dp[dAθpc : Rpcs(POIo(t),t)] = (-1)*Rocs(POIo)*sin(Aθoc(POIo))*sin(Aθpc(POIo(t),t))^(-2)*cos(Aθpc(POIo(t),t)) Subbing (2*) into (2) : = 1/5/Rocs(POIo)* { sin(Aθpc(POIo(t),t))^(0)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) + sin(Aθpc(POIo(t),t))^(1)*(-4)*Rpcs(POIo(t),t)^(-5)*(-1)*Rocs(POIo)*sin(Aθoc(POIo))*sin(Aθpc(POIo(t),t))^(-2)*cos(Aθpc(POIo(t),t)) } = 1/5/Rocs(POIo)* { cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) + (-4)*(-1)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(-1)*Rpcs(POIo(t),t)^(-5)*Rocs(POIo)*sin(Aθoc(POIo)) } Factor out = 1/5 /Rocs(POIo)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5)* { Rpcs(POIo(t),t) + 4 *sin(Aθpc(POIo(t),t))^(-1)*Rocs(POIo)*sin(Aθoc(POIo)) } = 1/5 /Rocs(POIo)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5)* { Rpcs(POIo(t),t) + 4*sin(Aθpc(POIo(t),t))^(-1)*Rocs(POIo)*sin(Aθoc(POIo)) } From "Rocv(POIo), Aθoc(POIo), Aφoc(POIo) are constants" : 5*) Rpcs (POIo(t),t)*sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) At t=0, the observer & particle reference frames coincide, and 3a) sin(Aθoc(POIo))|t=0 = sin(Aθpc(POIo(t),t))^(-1)|t=0 3b) Rocs(POIo) |t=0 = Rpcs(POIo(t),t) = 1/5/Rocs(POIo) *cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5)*{ Rpcs(POIo(t),t) + 4*Rocs(POIo) } = 1/5/Rocs(POIo) *cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5)*Rocs(POIo)*{ 1 + 4 } = { 1/5/Rocs(POIo)*Rocs(POIo)*5 } *cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5) = cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5) Summarizing : (3) dp[dAθpc : sin(Aθpc(POIo(t),t))^(1)*Rpcs(POIo(t),t)^(-4)/5/Rocs(POIo) ] = cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5) /*Which is the same as the integral term of (1), so the solution is correct. /*_endCmd /********************************************** >>> III. Derivations for a POIo = POIp(t) fixed in the observer reference frame (RFo) /***************************************** >>>>>> Basic measures /********************* >>>>>>>>> Figure "Basic measures for the observer reference frame RFo, using POIo=POIp(tx)" http://www.BillHowell.ca/ /media/bill/SWAPPER/Lucas - Universal Force/Images/Howell - Chapter 4 - POIo basic - cropped.png Reminders for Chapter 4 : Vonv(PART) = constant [Particle, observer] frames of reference (RFp) & (RFo) are IDENTICAL [scale, rotation, etc] at time t=0, apart from the given that the particle's reference frame (RFp) moves with velocity Vonv(PART) relative to RFo. Reminders for (POIo) - Point Of Interest that is FIXED in the observer reference frame (RFo) : [Roc,Ooc,Poc] and their derived concepts, are NOT functions of time, i.e. (POIp). Their derivatives with respect to time are zero. [Rpc,Opc,Ppc], their derived concepts, and their derivatives are all functions of time, i.e. (POIp) OBJECTIVES : Where convenient, express variables as functions of FIXED variables : (POIp) -> -Vodv(PART),t,Rpcv(POIp),Aθpc(POIp),Aφpc(POIp) (POIo) -> Vodv(PART),t,Rocv(POIo),Aθoc(POIo),Aφoc(POIo) /********************* >>>>>>>>> Rocv(POIo), Aθoc(POIo), Aφoc(POIo) are constants in RFo As per the introduction to this section, : This section deals with (POIo) - Points Of Interest(POI) which are fixed with respect to the observer reference frame (RFo). In other words, the (POIo) is stationary with respect to the observer and moves with constant relative velocity -Vonv(PART) with respect to the particle. /% (mathH)/* at time t when POIo and POIp(t) are the same point /% Rocv(POIo) = constant ≠ Rpcv(POIo(t),t) (mathH)/* at time t when POIo and POIp(t) are the same point /% Aθoc(POIo) = constant ≠ Aθpc(POIo(t),t) (endMath) (mathH)/* at time t when POIo and POIp(t) are the same point /% Aφoc(POIo) = constant ≠ Aφpc(POIo(t),t) (endMath) /********************* >>>>>>>>> Rocv(POIp(t),t) /*+-----+ Galilean transformation : /% (mathH) Rocv(POIp(t),t) = Rpcv(POIp) + Vonv(PART)*t (endMath) /*Limit checks : Dimensional consistency - OK, as all terms reduce to (length). /********************* >>>>>>>>> Rocs(POIp(t),t) Distance Rocs(POIp(t),t) from the (RFo) origin : /% Rocs(POIp(t),t) = |Rpcv(POIp) + Vonv(PART)*t| = { [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 }^(1/2) = { [Rpcs(POIp)*cos(Aθpc(POIo(t),t))]^2 + 2*Rpcs(POIp)*cos(Aθpc(POIp)*Vons(PART)*t + [Vons(PART)*t]^2 + [Rpcs(POIp)*sin(Aθpc(POIp))]^2 }^(1/2) = { Rpcs(POIp)^2 *[ cos(Aθpc(POIo(t),t))^2 + sin(Aθpc(POIp))^2 ] + 2*Rpcs(POIp)*cos(Aθpc(POIp)*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /*as sin^2 + cos^2 = 1 : /%(2) (mathH) Rocs(POIp(t),t) = { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) (endMath) /*Limit checks : Dimensional consistency - OK, as all terms reduce to (length). /********************* >>>>>>>>> RO0ocs(POIp(t),t) See Figure "Basic measures for a POIp". Distance of RO0ocs(POIp(t),t) from the (RFo) origin in O0ch direction (i.e. along L(PART)) : /% (mathH) Rθ0ocs(POIp(t),t) = Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t (endMath) /*Limit checks : Dimensional consistency - OK, as all terms reduce to (length). /********************* >>>>>>>>> ROPI2ocs(POIp(t),t) = constant See Figure "Basic measures for a POIp". Notice that the distance of ROPI2pcs(POIp) = ROPI2ocs(POIp(t),t) from L(PART) in Ppch=Poch direction (i.e. perpendicular to L(PART)) is a CONSTANT : /% ROPI2ocs(POIp(t),t) = Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t)) = Rpcs(POIp)*sin(Aθpc(POIp)) (mathH) ROPI2ocs(POIp(t),t) = Rpcs(POIp)*sin(Aθpc(POIp)) (endMath) /*Limit checks : Dimensional consistency - OK, as all terms reduce to (length). /********************* >>>>>>>>> sin(Aθoc(POIp(t),t)) From (3)&(2) : /% sin(Aθoc(POIp(t),t)) = Rpcs(POIp)*sin(Aθpc(POIp)) / Rocs(POIp(t),t) = Rpcs(POIp)*sin(Aθpc(POIp)) /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) (mathH) sin(Aθoc(POIp(t),t)) = Rpcs(POIp)*sin(Aθpc(POIp)) /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) (endMath) /*Limit checks : Dimensional consistency - OK, as all terms reduce to (dimensionless). /********************* >>>>>>>>> cos(Aθoc(POIp(t),t)) /% cos(Aθoc(POIp(t),t)) = Rθ0ocs(POIp(t),t) / Rocs(POIp(t),t) Subbing (5)&(2) : cos(Aθoc(POIp(t),t)) = [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] /{Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) (mathH) cos(Aθoc(POIp(t),t)) = [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / {Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) (endMath) /*Limit checks : Dimensional consistency - OK, as all terms reduce to (dimensionless). Limit check sin^2 + cos^2 = 1 /% sin(Aθoc(POIp(t),t))^2 + cos(Aθoc(POIp(t),t)) = [ Rpcs(POIp)*sin(Aθpc(POIp)) /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ]^2 +[ [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ]^2 = { [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ]^2 +[ Rpcs(POIp)*sin(Aθpc(POIp)) ]^2 } /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } = { [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) ]^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t +[ Vons(PART)*t ]^2 +[ Rpcs(POIp)*sin(Aθpc(POIp)) ]^2 } /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } = { Rpcs(POIp)^2*[cos(Aθpc(POIo(t),t))^2 + sin(Aθpc(POIp))^2 ] + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t +[ Vons(PART)*t ]^2 } /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } = { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t +[ Vons(PART)*t ]^2 } /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } = { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } = 1 /*OK - this agrees with earlier result (1) /********************* >>>>>>>>> R_O0_ocs(POIo) (mathH) R_O0_ocs(POIo) = Rocs(POIo)*cos(Aθocs(POI)) (endMath) ... need to do R_O0_ocs(POIp(t),t) /********************* >>>>>>>>> E0odv(POIo,t) = E0pdv(POIo(t),t) ≠ E0pdv(POIp) = constant, except when t = tx The electrostatic field at a fixed point POIo in observer space RFo is a function of time. The reference frame does NOT affect the measured electroSTATIC field at a point at the instant "t = tx" when POIo and POIp coincide, i.e. it is the same as seen from the particle frame of reference RFp as it is for the observer frame RFo (or any) frame at that instant in time, at a common point in space. This CONTRASTS to the INDUCED electric field EI, as discussed in a section3 below. Furthermore, for this work time delays for fields are ignored, such that the field MOVES WITH THE PARTICLE INSTANTANEOUSLY. /*+-----+ (RFp) basis Gauss's Law for a single point charge, in the particle reference frame (RFp) : /% (mathH) E0odv(POIo,t) = E0pdv(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) (endMath) /*+-----+ (RFo) basis /*From "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)" : /% (2) Rpcs(POIo(t),t) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /*Delete as not used here : /% (6) cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) /*Subbing (2)* & (6)* into (1)* : /%(3) E0odv(POIo,t) = E0pdv(POIp) = Q(PART)/Rpcs(POIp)^2*Rpch(POIp) = Q(PART) *Rpch(POIp) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^2 = Q(PART) *Rpch(POIp) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } Finally : (mathH) E0odv(POIo,t) = E0pdv(POIo(t),t) = Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } (endMath) /*As another different description of the unit displacement vector : Rpch(POIp(t),t) => Rodh(POIo) = unit-length displacement vector [start : RFo origin; end : POIo; length : 1] or [apex : ; theta : arccos(Aθpc(POIo(t),t)), phi : Aφpc] Note that : /% Rocs(POIo)*cos(Aθoc(POIo)) = Rθ0ocs(POIo) /*+--+ LIMIT CHECKS : Dimensionality check : OK as all units reduce to (charge/length^2) Later .... Dimensional consistency in SI units with : mu - magnetic permittivity epsilon - electric permeability Other checks? /********************* >>>>>>>>> E0ods(POIo,t) = E0pds(POIo(t),t) ≠ (for t ≠ tx) E0pds(POIp) = constant /%At t = tx, then POIo & POIp=POIo(tx) are coincident, such that E0ods(POIo(t)x) = E0pds(POIo(tx),tx) = E0pds(POIp) = #constant. Note that A formal derivation is needed (or just refer to Lucas). /*+-----+ (RFp) basis /% (mathH) E0ods(POIo,t) = |Q(PART)|/Rpcs(POIo(t),t)^2 (endMath) /*+-----+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (charge/length^2). (ignoring electric permeability for Gaussian coordinates) /*+-----+ (RFo) basis From "E0odv(POIo,t) = E0pdv(POIo(t),t)" : /% (5)* E0odv(POIo,t) = Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } /*Subbing (5)* into (2) : /% E0ods(POIo,t) = |E0odv(POIo)| = | Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } | = |Q(PART)|/{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } Summarizing : (mathH) E0ods(POIo,t) = |Q(PART)|/{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } (endMath) /*Key point from "Scalar absolute values, [vector, matrix] norms - simplification of expressions" : - EIpdv(POIo(t),t) - is in the opposite direction to E0pdv(POIo(t),t) vector norms - if some terms are vectors - multiplicative/divisive : |product(x1,x2,x3,...)| = product(|x1|,|x2|,|x3|,...) - subtraction/ addition : |sum (x1,x2,x3,...)| ≠ ≠ product(|x1|,|x2|,|x3|,...) in general (although it may be true in some cases) /********************* >>>>>>>>> Figure "BTodv(POIo,t)" http://www.BillHowell.ca/ /media/bill/SWAPPER/Lucas - Universal Force/Images/Howell - Chapter 4 - BTodv(POIo,t) - cropped.png /********************* >>>>>>>>> BIodv(POIo,t) = BIodv(POIp(t),t) ≠ BIpdv(POIp) = 0 Induced magnetic fields DO arise for Points Of Interest (POIo) that are fixed in the observer reference frame (RFo), as there is a relative velocity between the POIo and the particle, and therefore between POIo and ETodv(POIp(t),t), which in turn is the sum of E0odv(POIp(t),t) and EIodv(POIp(t),t)] (superposition applies - linear system). There is a recurrent nature to the derivation of an expression, as EIodv(POIp(t),t) itself depends on changes in the total magnetic field. This is covered in Section III "Chapter 4 - Derivations for a POI fixed in the observer reference frame (RFo)". /********************* >>>>>>>>> BTodv(POIo,t) = BTodv(POIp(t),t) ≠ BTpdv(POIp) = BTpdv(POIo(t),t) = 0 ???? Beginning with : /% BTodv(POIo,t) = B0odv(POIo,t) + BIodv(POIo,t) /*where B0odv(POIo,t) - magnetic field external (currents, permanent mags) in (RFo) BIodv(POIo,t) - magnetic field induced by charge Q(PART), which moves in RFo but does NOT move in (RFp) Later : This is addressed in "Howell - math of Lucas Universal Force.ndf", and further below where Lenz'sLaw and the Thomas Barnes iterations are addressed. /********************* >>>>>>>>> Prediction of direction of field (B), given that the current I flows in the direction of the thum Run command to see $ eog "$d_images""Wikipedia - Right-hand rule for B from E.png" & Prediction of direction of field (B), given that the current I flows in the direction of the thumb https://en.wikipedia.org/wiki/Right-hand_rule In order to compare to Lucas's intermediate results, it is handy to have an expression BEFORE invoking of Lenz's Law, and before Lucas's results from Thomas Barnes iterations. NOTE : How does Lucas's comment affect what is below? ???"... only static E fields give rise to B ..."??? p67h0.3 Equation (4-13) "... Another significant aspect of this work is that the induced B X v field is not electrostatic in nature. According to Cullwick [8], Hooper [10], and Moon and Spencer [12], this means that the linear superposition principle as applied to electric fields does not hold for the B X v generated fields. Thus in electrodynamics one must explicitly keep track of both electrostatic fields and the induced fields. Using the basic equations of electrodynamics one must calculate explicitly the induced fields in order to obtain the total fields of the moving charged particle. (Note that the covariant form of electrodynamics based upon Maxwell's equations assumes that the superposition principle holds and does not treat electrostatic and induced fields separately in disagreement with the experimental results cited. [10]) ..." (I wonder if he meant "E X v"?) BUT : equation (4-6) Lorentz force has v X B equation (4-11a) is a linear superposition of E0 & EI!!! equation (4-13) IS a linear supoerposition of E fields for calculating BI!!! WRONG!!! - While the EIo calculation is iterative (previous sub-subsection), the resulting EIo result is NOT used to calculate BIo, only the E0o portion applies. ??????? ????????? WRONG ??? : Assuming : /% BTpdv(POIo(t),t) = B0pdv(POIo(t),t) + (BIpdv(POIo(t),t) = 0) 27Aug2019 no BIpdv?? /*where B0pdv(POIo(t),t) - magnetic field external (currents, permanent mags) in (RFo) BIpdv(POIo(t),t) - magnetic field induced by the moving charge Q(PART) in (RFo) For the Chapter 4 situation, there is no "external" magnetic field (i.e. from other sources, independent of the particle), and so B0 is a constant zero (assuming that this is the definition of B0). Induced magnetic fields BIodv(POIo,t) do arise for Points Of Interest (POIo) that are fixed in the observer reference frame (RFo), as there is a relative velocity between the POIo and the particle (and therefore between POIo and E0ocv(POIo(t),t)). Points Of Interest (POIp) that are fixed in the particle reference frame (RFp) are covered in the previous section II "Chapter 4 - Derivations for a POIp fixed in the particle reference frame (RFp)". Here there is NO induced magnetic field as there is NO relative velocity between the POIp and the charged particle (and therefore between POIp and E0pcv(POIp)). /%From "E0odv(POIo,t) = E0pdv(POIo(t),t)" : (1) E0odv(POIo,t) = E0pdv(POIo(t),t) = Q(PART)/Rpcv(POIo(t),t)^2 and EIpdv(POIo(t),t) = EIodv(POIo,t) /* ASSUMING that E0odv(POIo,t) & EIodv(POIo,t) are in the same direction!!! Note : Lenz's Law (Lucas p64h0.5 Eq (4-5), p70h0.9 Eq (4-31)) provides assurance that E0odv(POIo,t) & EIodv(POIo,t) are in the same direction ... ??? 19Dec2017 NO! They are in OPPOSITE directions!!! This is important for scalar measures. /%From "ETodv(POIo,t)= ((ETpdv(POIo(t),t)=E0pdv(POIp))=E0pdv(POIp))" : (1) E0pdv(POIo(t),t) = E0pdv(POIo(t),t) + EIpdv(POIo(t),t) = E0odv(POIo,t) = E0odv(POIo,t) + EIodv(POIo,t) /* /% (mathH)/* Generalized Ampere's Law : /% BIodv(POIo,t) = Vonv(PART)/c X EOpdv(POIo(t),t) (endMath) /*From Lucas p67h0.6 Eqn (4-13) : /% (mathH)/* (4-13) Generalized Ampere's Law : /% BTpdv(POIo(t),t) = Vonv(PART)/c X [E0pdv(POIo(t),t) + EIpdv(POIo(t),t)] (endMath) /* ("X" is the vector cross-product operator) : /*+-----+ (RFp) basis From "E0odv(POIo,t) = E0pdv(POIo(t),t)", subbing (1)* into (4-13)* : /%(1) BTpdv(POIo(t),t) = BTodv(POIo,t) = Vonv(PART)/c X [ E0pdv(POIo(t),t) + EIpdv(POIo(t),t) ] = Vonv(PART)/c X Q(PART)/Rpcv(POIo(t),t)^2 + Vonv(PART)/c X EIpdv(POIo(t),t) /*Key point!! Now, when using scalars, it must be kept in mine that : - EIpdv(POIo(t),t) - is in the opposite direction to E0pdv(POIo(t),t) - from "Scalar absolute values, [vector, matrix] norms - simplification of expressions" vector norms - if some terms are vectors - multiplicative/divisive : |product(x1,x2,x3,...)| = product(|x1|,|x2|,|x3|,...) - subtraction/ addition : |sum (x1,x2,x3,...)| ≠ ≠ product(|x1|,|x2|,|x3|,...) in general (although it may be true in some cases) Therefore, EIpds(POIo(t),t) must be multiplied by -1 when added to E0pds(POIo(t),t) !!! /% = Vons(PART)/c *Q(PART)/Rpcs(POIo(t),t)^2 *sin[Aθpd(Vonv,Rpch(POIo(t),t))]*Rodh(Vonv_X_Rpcv(POIo(t),t)) - Vons(PART)/c *EIpds(POIo(t),t) *sin[Aθpd(Vonv,Rpch(POIo(t),t))]*Rodh(Vonv_X_Rpcv(POIo(t),t)) = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t) ] where : (2) Aθpd(Vonv,Rpch(POIo(t),t)) is the Aθ (theta) angle between the Vonv(PART) & E vectors (3) Aφpd(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction, which IS the direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) (4) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Note that it is a pure RFo unit vector, constant with time Notes : Aφpd(Vonv_X_Rpcv(POIo(t),t)) is the direction perpendicular to Voch X Rpch(POIo(t),t) using the right-hand rule. /*But : as this doesn't change in the Chapter 4 derivations it isn't shown in the formulae I haven't properly accounted for the signs yet (right hand rule). Notice that, because Vonh is the same as the O0pch = O0och coordinate direction for both (RFo) and (RFp) (i.e. it is parallel to the particle motion in (RFo)), the angle of intersection is simply Aθpc(POIo(t),t). In other words : /%(5) Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) /*Therefore : /% (mathH) BTpdv(POIo(t),t) = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t)] (endMath) where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθoc(POIo) (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφoc(POIo) (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Aφpd(Vonv_X_Rpcv(POIo(t),t)) & Rodh(Vonv_X_Rpcv(POIo(t),t)) are CONSTANT for Chapter 4 /*which is equivalent to Lucas Eqn (4-15) CAUTION!!! : ??? Lucas p67h0.3 - "B X v" field is not electrostatic in nature : what does this mean? Is this an error, should it be "E X v"???? /*+--+ Limit checks : Dimensional consistency, noting that unit vectors are taken as dimensionless : /% (6) BTpdv(POIo(t),t) = BTodv(POIo,t) = Vons(PART)/c *sin(Aθpc(POIo(t),t)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART)/Rpcs(POIo(t),t)^2 + EIpds(POIo(t),t) ] = (length/time)/(length/time) *(dimensionless) *(dimensionless) *[ (charge) /(length^2) + (charge/length^2) ] = (charge/length^2) /*OK as all units reduce to (charge/length^2) (ignoring permittivity & permeability for Gaussian units) /*+-----+ (RFo) basis Starting with the expression above in (RFp) coordinates, and noting that BTodv(POIo,t) = BTpdv(POIo(t),t) : /% (6) BTpdv(POIo(t),t) = BTodv(POIo,t) = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t) ] /*From "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)" : /% (2) Rpcs(POIo(t),t)) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) (5) sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /*Finally, noting that, as RFo and RFp are aligned, but with offset along the vov direction : (7) Rodh(Vonv_X_Rpcv(POIo(t),t)) = Rodh(Vonv_X_Rpcv(POIo)) NOTE: EIpdv(POIo(t),t) = EIodv(POIo,t) ASSUMING that E0odv(POIo,t) & EIodv(POIo,t) are in the same direction!!! Note : Lenz's Law (Lucas p64h0.5 Eq (4-5), p70h0.9 Eq (4-31)) provides assurance that E0odv(POIo,t) & EIodv(POIo,t) are in the same direction ... ??? /*Subbing (2)*, (6)*, and (7) into (6) : /%(8) BTpdv(POIo(t),t) = BTodv(POIo,t) = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART)/Rpcs(POIo(t),t)^2 - EIods(POIo(t),t) ] = Vons(PART)/c *Rodh(Vonv_X_Rpcv(POIo(t),t)) *sin(Aθpc(POIo(t),t)) *[ Q(PART) /Rpcs(POIo(t),t)^2 - EIods(POIo(t),t) ] = Vons(PART)/c *Rodh(Vonv_X_Rpcv(POIo(t),t)) * Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) *[ Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^2 - EIods(POIo(t),t) ] = Vons(PART)/c *Rocs(POIo)*sin(Aθoc(POIo)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) - EIods(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ] /*What is the geometric interpretation of ? : /% Rocs(POIo) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) = Rocs(POIo)/Rpcs(POIo(t),t)^3 Rocs(POIo) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) = Rocs(POIo)/Rpcs(POIo(t),t) /*Hmm... should be able to do something with this. Repeating the result : /% (mathH) BTodv(POIo,t) = Vons(PART)/c*Rocs(POIo)*sin(Aθoc(POIo))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) - EIods(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ] (endMath) where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) /*+--+ LIMIT CHECKS : Dimensional consistency, noting that unit vectors are taken as dimensionless : /% (9) BTpdv(POIo(t),t) = BTodv(POIo,t) = Vons(PART)/c *Rocs(POIo)*sin(Aθoc(POIo)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) + EIods(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ] = (length/time)/(length/time) *(length) *(dimensionless) *(dimensionless) *[ (charge) /(length^3) + (charge/length^2)/(length) ] = (length) *[ (charge/length^3) + (charge/length^3) ] = (charge/length^2) /*OK as all units reduce to (charge/length^2) (ignoring permittivity & permeability for Gaussian units) /********************* >>>>>>>>> EIodv(POIo,t) = EIodv(POIp(t),t) ≠ EIpdv(POIp) = 0 Again, the INDUCED electric field arises from CHANGES in the magnetic field at a point. In the observer reference frame RFo, the electric field at POIo = POIp(t) (that is, POIp at time t) DOES change, and gives rise to an induced field (which is derived in a later section). Later : This is addressed in "Howell - math of Lucas Universal Force.ndf", and further below where Lenz's Law and the Thomas Barnes iterations are addressed. /********************* >>>>>>>>> ETodv(POIo,t) = ETodv(POIp(t),t) = E0odv(POIp(t),t) + EIodv(POIp(t),t) ≠ ETpdv(POIp) = 0 Later : This is addressed in "Howell - math of Lucas Universal Force.ndf", and further below where Lenz'sLaw and the Thomas Barnes iterations are addressed. /% (mathH) EIods(POIo,t) = EIpds(POIo(t),t) ≠ EIpds(POIp) = 0 (endMath) /*For the "Basic math & calculus", [EIodv(POIo,t) = EIpdv(POIo(t),t), EIods(POIo(t),t) = EIpds(POIo(t),t)] (and their derivatives later) are being left "as is" in the expression. In later sections, Lenz's Induction Law and Thomas Barnes iterations will yield closed expressions for these variables. Key point from "Scalar absolute values, [vector, matrix] norms - simplification of expressions" : - EIpdv(POIo(t),t) - is in the opposite direction to E0pdv(POIo(t),t) vector norms - if some terms are vectors - multiplicative/divisive : |product(x1,x2,x3,...)| = product(|x1|,|x2|,|x3|,...) - subtraction/ addition : |sum (x1,x2,x3,...)| ≠ ≠ product(|x1|,|x2|,|x3|,...) in general (although it may be true in some cases) /********************* >>>>>>>>> ETodv(POIo,t) = ETpdv(POIo(t),t) Following Lucas p67h0.6 Eqn (4-13) & (4-41) : /% (1)* ((ETpdv(POIo(t),t)=E0pdv(POIp))=E0pdv(POIp)) = E0pdv(POIo(t),t) + EIpdv(POIo(t),t) /*From "" : Following Lucas p67h0.6 Eqn (4-13) & (4-41) : /% (2)* = ETodv(POIo,t) = E0odv(POIo,t) + EIodv(POIo,t) /********************* /%>>>>>>>>> ETods(POIo,t) = ((ETpds(POIo(t),t)=E0pds(POIp))=E0pds(POIp)) (mathH)/* Following Lucas p67h0.6 Eqn (4-13) & (4-41) /% E0pdv(POIo(t),t) + EIpdv(POIo(t),t) = ETodv(POIo,t) = E0odv(POIo,t) + EIodv(POIo,t) (endMath) /*For scalar calculations, must keep in mind Lenz's Law (induced is opposite direction of "inducer") Key point!! Now, when using scalars, it must be kept in mine that : - EIpdv(POIo(t),t) - is in the opposite direction to E0pdv(POIo(t),t) - from "Scalar absolute values, [vector, matrix] norms - simplification of expressions" 19Dec2017 OK - I get it now after coming back after ~1 year. This is required for the SCALAR equations! vector norms - if some terms are vectors - multiplicative/divisive : |product(x1,x2,x3,...)| = product(|x1|,|x2|,|x3|,...) - subtraction/ addition : |sum (x1,x2,x3,...)| ≠ ≠ product(|x1|,|x2|,|x3|,...) in general (although it may be true in some cases) Therefore : /% (1)* ETpds(POIo(t),t) = E0pds(POIo(t),t) - EIpds(POIo(t),t) (2)* = ETods(POIo,t) = E0ods(POIo,t) - EIods(POIo,t) /*+-----+ (RFp) basis From "E0ods(POIo,t) = E0pds(POIo(t),t)" : /% (mathH) E0pds(POIo(t),t) = Q(PART)/Rpcs(POIp)^2 (endMath) /*Subbing (1)* into (1) : ETpds(POIo(t),t) = E0pds(POIo(t),t) - EIpds(POIo(t),t) = Q(PART)/Rpcs(POIp(t),t)^2 - EIpds(POIo(t),t) (mathH)/* seems wrong???? /% ETpds(POIo(t),t) = Q(PART)/Rpcs(POIp(t),t)^2 - EIpds(POIo(t),t) (endMath) /*+-----+ (RFo) basis /%From "E0ods(POIo,t) = E0pds(POIo(t),t)" : (2)** E0ods(POIo) = |Q(PART)|/{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } Subbing (2)** into (2)* (2)* ETods(POIo(t),t) = E0ods(POIo,t) - EIods(POIo(t),t) So : (4) ETods(POIo(t),t) = |Q(PART)|/{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } - EIods(POIo(t),t) /***************************************** >>>>>> Derivatives /********************* >>>>>>>>> Figure "Calculus for RFo, using POIo=POIp(t)" http://www.BillHowell.ca/ /media/bill/SWAPPER/Lucas - Universal Force/Images/Howell - Chapter 4 - POIp calculus - cropped.png Reminders for Chapter 4 : Vonv(PART) = constant [Particle, observer] frames of reference (RFp) & (RFo) are IDENTICAL [scale, rotation, etc] at time t=0, apart from the given that the particle's reference frame (RFp) moves with velocity v relative to RFo. Reminders for (POIp) - Point Of Interest that is FIXED in the particle reference frame (RFp) : [Rpc,Opc,Ppc] and their derived concepts, are NOT functions of time, i.e. (POIp). Their derivatives with respect to time are zero. [Roc,Ooc,Poc], their related concepts, and their derivatives are all functions of time, i.e. (POIp(t),t) OBJECTIVES : Where convenient, express variables as functions of FIXED variables : (POIp) -> -Vodv(PART),t,Rpcv(POIp),Aθpc(POIp),Aφpc(POIp) (POIo) -> Vodv(PART),t,Rocv(POIo),Aθoc(POIo),Aφoc(POIo) /********************* >>>>>>>>> ∂[∂(t): Rocv(POIo)], ∂[∂(t): Aθoc(POIo)], ∂[∂(t): Aφoc(POIo)] = 0 As per "Rocv(POIo), Aθoc(POIo), Aφoc(POIo)" above, Roc and Ooc are constants for a given POIo, so their derivatives are zero : (mathH) ∂[∂(t): Rocv(POIo)] = 0 (endMath) (mathH) ∂[∂(t): Aθoc(POIo)] = 0 (endMath) (mathH) ∂[∂(t): Aφoc(POIo)] = 0 (endMath) /********************* >>>>>>>>> ∂[∂(t): Rocv(POIp(t),t) ] From Galilaean invariance rov - Vonv(PART)*t = Rpcv(POIp) = constant Also, for any POI in RFo (other than the particle itself), rov is a constant (fixed position for Chapter 4 - there are no other "pieces" moving with respect to RFo. Therefore /% ∂[∂(t): Rocv(POIp(t),t)] = ∂[∂(t): Rpcv(POIp) + Vonv(PART)*t] = ∂[∂(t): Rpcv(POIp)] + ∂[∂(t): Vonv(PART)*t] = 0 + Vonv(PART) (mathH) ∂[∂(t): Rocv(POIp(t),t) = Vonv(PART)] (endMath) /*This is expected. /*+-----+ Limit checks : Dimensional consistency - OK, as all terms reduce to (length/time). /********************* /%>>>>>>>>> ∂[∂(t): Rocs(POIp(t),t)] /*???Check this - improper derivative (use vector approach!!) /% Rocs(POIp(t),t) = |Rocv(POIp(t),t)| See "∂[∂(t): Rocv(POIp(t),t)] " above : ∂[∂(t): Rocs(POIp(t),t)] = ∂[∂(t): |Rocv(POIp(t),t)|] /*Kahan formulation : d||z|| = u_T dotPRod dz / ||z|| where u_T is the linear functional dual to z wrt ||...|| HERE I INTERPRET : u_T = Rocv(POIp(t),t)], so : /% ∂[∂(t): Rocs(POIp(t),t)] = ∂[∂(t): |Roc(POIp(t),t)|] = Rocv(POIp(t),t) dotPRod ∂[∂(t): Rocv(POIp(t),t)] / |Rocv(POIp(t),t)| from Eq (1) in "∂[∂(t): Rocv(POIp(t),t)]" above : ∂[∂(t): Rocv(POIp(t),t)] = Vonv(PART) = Rocv(POIp(t),t) dotPRod Vonv(PART) / |Rocv(POIp(t),t)| = |Rocv(POIp(t),t)|*cosAθoc(POIp(t),t)*|Vonv(PART)| / |Rocv(POIp(t),t)| = Vons(PART) *cosAθoc(POIp(t),t) (mathH) ∂[∂(t): Rocs(POIp(t),t)] = Vons(PART)*cosAθoc(POIp(t),t) (endMath) /*Want in terms of variables of (POIp) : /*+-----+ CONFIRMATION : Show Figure "∂[∂(t): Rocs(POIp(t),t)] explained" /%(2) |Rocv(POIp(t),t)| = Rocs(POIp(t),t) = { [(Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))]^2 + [Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]^2 }^(1/2) (3) ∂[∂(t): Rocs(POIp(t),t)] = ∂[∂(t): |Rocv(POIp(t),t)|] = 1/2*{ [(Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))]^2 + [Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]^2 }^(-1/2) *∂[∂(t): {(Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))]^2 + [Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]^2] (4) ∂[∂(t): [(Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))]^2 + [Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]^2] = ∂[∂(t): [(Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))]^2] + ∂[∂(t): Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]^2 ] But for all t, the following are constants, as Vodv(POI) is parallel to the coordinate axis [Rθ0och, Rθ0pch] : Rpcs(POIp)*sin(Aθpc(POIp)) = Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t)) = constant over t for any (POIo), (POIp) therefore ∂[∂(t): [(Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))]^2 + [Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]^2] = ∂[∂(t): [Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]^2] = 2 *[Rocs(POIp(t),t)*cosAθoc(POIp(t),t)] *∂[∂(t): Rocs(POIp(t),t)*cosAθoc(POIp(t),t)] Where from "∂[∂(t): Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]" below : ∂[∂(t): Rocs(POIp(t),t)*cosAθoc(POIp(t),t)] = Vons(PART) therefore (5) ∂[∂(t): [(Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))]^2 + [Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]^2] = 2*Vons(PART)*Rocs(POIp(t),t)*cosAθoc(POIp(t),t) Repeating (3) ∂[∂(t): Rocs(POIp(t),t)] = ∂[∂(t): |Rocv(POIp(t),t)|] = 1/2*{ [(Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))]^2 + [Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]^2 }^(-1/2) *∂[∂(t): {(Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))]^2 + [Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]^2] Substituting (5) into (3) : = 1/2*{ [(Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))]^2 + [Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]^2 }^(-1/2) *2*Vons(PART) * Rocs(POIp(t),t)*cosAθoc(POIp(t),t) = Vons(PART) * Rocs(POIp(t),t)*cosAθoc(POIp(t),t) *{ [(Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))]^2 + [Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]^2 }^(-1/2) = Vons(PART) * Rocs(POIp(t),t)*cosAθoc(POIp(t),t) /Rocs(POIp(t),t)*{sin(Aθoc(POIp(t),t))^2 + cosAθoc(POIp(t),t) ^2 }^(-1/2) But 1 = { sin(Aθoc(POIp(t),t))^2 + cosAθoc(POIp(t),t)^2 }^(-1/2) therefore : (6) ∂[∂(t): Rocs(POIp(t),t)] = Vons(PART)*cosAθoc(POIp(t),t) /*This is the same as (1) above, confirming that result. /*+-----+ OBJECTIVES : Where convenient, express variables as functions of FIXED variables : (POIp) -> -Vodv(PART),t,Rpcv(POIp),Aθpc(POIp),Aφpc(POIp) (POIo) -> Vodv(PART),t,Rocv(POIo),Aθoc(POIo),Aφoc(POIo) /*From "Relating [Rocv,Rocs,ROPI2ocs ,rO0ocs,sin(Aθoc),cos(Aθoc)]@(POIp(t),t) to [...]@(POIp)" : /% (2) Rocs(POIp(t),t)) = { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIp)*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) so : Rocs(POIp(t),t) = { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cosAθpc(POIp)*Vons(PART)*t + (Vons(PART)*t)^2 }^( 1/2) and : ∂[∂(t): Rocs(POIp(t),t)] = (1/2)*{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cosAθpc(POIp)*Vons(PART)*t + (Vons(PART)*t)^2 }^(-1/2) * ( 0 + 2*Rpcs(POIp)*cosAθpc(POIp)*Vons(PART) + 2*(Vons(PART)*t)*Vons(PART) } = (1/2)*{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cosAθpc(POIp)*Vons(PART)*t + (Vons(PART)*t)^2 }^(-1/2) * { 0 + 2*Rpcs(POIp)*cosAθpc(POIp)*Vons(PART) + 2 *(Vons(PART)*t)*Vons(PART) } = { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cosAθpc(POIp)*Vons(PART)*t + (Vons(PART)*t)^2 }^(-1/2) *{ Rpcs(POIp)*cosAθpc(POIp)*Vons(PART) + Vons(PART)^2*t } then : (mathH) ∂[∂(t): Rocs(POIp(t),t)] = {Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + Vons(PART)^2*t} / {Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + (Vons(PART)*t)^2 }^(1/2) (endMath) /*+-----+ Limit checks : Dimensional consistency - OK, as all terms reduce to (length/time). As t -> +- infinity : /% (7) ∂[∂(t): Rocs(POIp(t),t)] = ( alpha(POIp) + Vons(PART)^2*t } /{ Rpcs(POIp)^2 + 2*alpha(POIp)*t + (Vons(PART) *t)^2 }^(1/2) tends to : => Vons(PART)^2*t / { (Vons(PART)*t)^2 }^(1/2) = +- Vons(PART) which is expected. Limit check : As t -> +- 0 : (7) ∂[∂(t): Rocs(POIp(t),t)] = ( alpha(POIp) + Vons(PART)^2*t } /{ Rpcs(POIp)^2 + 2*alpha(POIp)*t + (Vons(PART) *t)^2 }^(1/2) = ( alpha(POIp) + 0 } /{ Rpcs(POIp)^2 + 0 + 0 }^(1/2) = alpha(POIp) / Rpcs(POIp) where alpha(POIp) = Rpcs(POIp)*cosAθpc(POIp)*Vons(PART) so : ∂[∂(t): Rocs(POIp(t),t)] = Rpcs(POIp)*cosAθpc(POIp)*Vons(PART) / Rpcs(POIp) = cosAθpc(POIp)*Vons(PART) /*Harder to tell this is OK... Limit check : /%As cosAθoc(POIp(t),t) -> 0 : Vons(PART)*t = -cosAθpc(POIp)*Rpcs(POIp) /********************* /%>>>>>>>>> ∂[∂(t): sin(Aθoc(POIp(t),t))] /*For Chapter 4, v = constant, [Particle, observer] frames of reference are IDENTICAL [scale, rotation, etc] at time t=0, apart from the given that the particle's reference frame (RFp) moves with velocity v relative to RFo. See "Howell - math of Lucas Universal Force.ndf" (4-17) 'Sperical coordinate transforms ' Figure "Chapter 4 reference frames" /% Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t)) = Rpcs(POIp)*sin(Aθpc(POIp)) sin(Aθoc(POIp(t),t)) = Rpcs(POIp)/Rocs(POIp(t),t)*sin(Aθpc(POIp)) but [Rpcs(POIp),sin(Aθpc(POIp))] are constants, therefore ∂[∂(t): sin(Aθoc(POIp(t),t))] = Rpcs(POIp)*sin(Aθpc(POIp)) * ∂[∂(t): 1/Rocs(POIp(t),t)] from "∂[∂(t): Rocs(POIp(t),t)] = ∂[∂(t): |Rocv(POIp(t),t)|]" : (1) ∂[∂(t): Rocs(POIp(t),t)] = Vons(PART)*cosAθoc(POIp(t),t) therefore (1) ∂[∂(t): sin(Aθoc(POIp(t),t))] = Rpcs(POIp)*sin(Aθpc(POIp))*(-1)/Rocs(POIp(t),t)^2*Vons(PART)*cosAθoc(POIp(t),t) = -Rpcs(POIp)*sin(Aθpc(POIp))*Vons(PART) *cosAθoc(POIp(t),t)/Rocs(POIp(t),t)^2 I want to express this all in the particle reference frame (RFp) -> (POIp), Vons(PART), t Again : Rpcs(POIp)*sin(Aθpc(POIp)) = Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t)) therefore (2) cosAθoc(POIp(t),t)/Rocs(POIp(t),t)^2 = cosAθoc(POIp(t),t)* [ sin(Aθoc(POIp(t),t))/Rpcs(POIp)/sin(Aθpc(POIp)) ]^2 /*From "Relating [Rocv,Rocs,ROPI2ocs ,rO0ocs,sin(Aθoc),cos(Aθoc)]@(POIp(t),t) to [...]@(POIp)" : /% (5) sin(Aθoc(POIp(t),t)) = Rpcs(POIp)*sin(Aθpc(POIp)) /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /*From "Relating [Rocv,Rocs,ROPI2ocs ,rO0ocs,sin(Aθoc),cos(Aθoc)]@(POIp(t),t) to [...]@(POIp)" : : /% (6) cos(Aθoc(POIp(t),t)) = [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] /{Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing into (2) : (3) cosAθoc(POIp(t),t)*sin(Aθoc(POIp(t),t))^2 = [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) * [ Rpcs(POIp)*sin(Aθpc(POIp)) ]^2 / { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } = [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] * [ Rpcs(POIp)*sin(Aθpc(POIp)) ]^2 / { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) Subbbing end result of (3) into (2) : (4) 2) cosAθoc(POIp(t),t)/Rocs(POIp(t),t)^2 = cosAθoc(POIp(t),t)* [ sin(Aθoc(POIp(t),t))/Rpcs(POIp)/sin(Aθpc(POIp)) ]^2 = [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] * [ Rpcs(POIp)*sin(Aθpc(POIp)) ]^2 / { [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 }^(3/2) / [ Rpcs(POIp)*sin(Aθpc(POIp)) ]^2 = [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / { [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 }^(3/2) Subbbing end result of (4) into (1) : (1) ∂[∂(t): sin(Aθoc(POIp(t),t))] = -Rpcs(POIp)*sin(Aθpc(POIp))*Vons(PART) *cosAθoc(POIp(t),t)/Rocs(POIp(t),t)^2 = -Rpcs(POIp)*sin(Aθpc(POIp)) *Vons(PART) [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / { [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 }^(3/2) (mathH) ∂[∂(t): sin(Aθoc(POIp(t),t))] = -Rpcs(POIp)*sin(Aθpc(POIp)) * Vons(PART) * [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / { [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 }^(3/2) (endMath) /*+-----+ ALTERNATIVE derivation /% (1) ∂[∂(t): sin(Aθoc(POIp(t),t))] = -Rpcs(POIp)*sin(Aθpc(POIp))*Vons(PART) *cosAθoc(POIp(t),t)/Rocs(POIp(t),t)^2 /*From "Relating [Rocv,Rocs,ROPI2ocs ,rO0ocs,sin(Aθoc),cos(Aθoc)]@(POIp(t),t) to [...]@(POIp)" : /% (2) Rocs(POIp(t),t)) = |Rpcv(POIp) + Vonv(PART)*t| = { [Rpcs(POIp)*cos(Aθpc(POIp)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 }^(1/2) /*From "Relating [Rocv,Rocs,ROPI2ocs ,rO0ocs,sin(Aθoc),cos(Aθoc)]@(POIp(t),t) to [...]@(POIp)" : : /% (6) cos(Aθoc(POIp(t),t)) = [ Rpcs(POIp)*cos(Aθpc(POIp)) + Vons(PART)*t ] / { [Rpcs(POIp)*cos(Aθpc(POIp)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 }^(1/2) so : ∂[∂(t): sin(Aθoc(POIp(t),t))] = -Rpcs(POIp)*sin(Aθpc(POIp)) *Vons(PART) *cosAθoc(POIp(t),t)/Rocs(POIp(t),t)^2 = -Rpcs(POIp)*sin(Aθpc(POIp)) *Vons(PART) * [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / { [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 }^(1/2) / { [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 } therefore : (6) ∂[∂(t): sin(Aθoc(POIp(t),t))] = -Rpcs(POIp)*sin(Aθpc(POIp)) * Vons(PART) * [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / { [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 }^(3/2) /*Which is the same as (5) above. Equation (5) = Equation (6), so at least this checks - but these aren't really independent. /*+-----+ Limit checks : Dimensional consistency - OK, as all terms reduce to (time)^(-1). As (6) = (5), take this as a confirmation of sorts. /%As cos(Aθpc(POIo(t),t)) -> 0 (therefore sin(Aθpc(POIp)) -> +- 1) /********************* /%>>>>>>>>> ∂[∂(t): cosAθoc(POIp(t),t)] /*Reminders for Chapter 4 : Vonv(PART) = constant [Particle, observer] frames of reference (RFp) & (RFo) are IDENTICAL [scale, rotation, etc] at time t=0, apart from the given that the particle's reference frame (RFp) moves with velocity Vonv(PART) relative to RFo. Reminders for (POIp) - Point Of Interest that is FIXED in the particle reference frame (RFp) : [Rpc,Opc,Ppc] and their derived concepts, are NOT functions of time, i.e. (POIp). Their derivatives with respect to time are zero. [Roc,Ooc,Poc], their derived concepts, and their derivatives are all functions of time, i.e. (POIp(t),t) OBJECTIVES : Where convenient, express variables as functions of FIXED variables : (POIp) -> -Vodv(PART),t,Rpcv(POIp),Aθpc(POIp),Aφpc(POIp) (POIo) -> Vodv(PART),t,Rocv(POIo),Aθoc(POIo),Aφoc(POIo) /*+-----+ Cosine expression /*From "Relating [Rocv,Rocs,ROPI2ocs ,rO0ocs,sin(Aθoc),cos(Aθoc)]@(POIp(t),t) to [...]@(POIp)" : Distance of RO0ocs(POIp(t),t) from the (RFo) origin in RO0och direction (i.e. along L(PART)) : /% (4) Rθ0ocs(POIp(t),t) = Rocs(POIp(t),t)*cos(Aθoc(POIp(t),t)) = Rpcs(POIp)*cos(Aθpc(POIp)) + Vons(PART)*t therefore : cos(Aθoc(POIp(t),t)) = [Rpcs(POIp)*cos(Aθpc(POIp)) + Vons(PART)*t ] / Rocs(POIp(t),t) /*From "Relating [Rocv,Rocs,ROPI2ocs ,rO0ocs,sin(Aθoc),cos(Aθoc)]@(POIp(t),t) to [...]@(POIp)" : /% (2) Rocs(POIp(t),t)) = { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) therefore : cos(Aθoc(POIp(t),t)) = [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / Rocs(POIp(t),t) = [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /*+-----+ Expression for ∂[∂(t): cosAθoc(POIp(t),t)] ∂[∂(t): cosAθoc(POIp(t),t)] = ∂[∂(t): [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ] = ∂[∂(t): [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] ] /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) + [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] *∂[∂(t):{Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-1/2)] = Vons(PART) /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) + [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] *{(-1/2)/{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) *∂[∂(t): { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } } = Vons(PART) /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) - [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] *{ /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) * (1/2)*{ 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + 2*[Vons(PART)*t] *∂[∂(t): [Vons(PART)*t] ] } } = Vons(PART) /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) - *[ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] *{1/{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) * { Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + [Vons(PART)^2*t] } } = Vons(PART) / { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) - [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] * { Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + [Vons(PART)^2*t] } / { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) = Vons(PART) /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) - *[ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] *{ + Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + [Vons(PART)^2*t] } /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) /*Leaving it at that : /% (mathH) ∂[∂(t): cosAθoc(POIp(t),t)] = Vons(PART) / {Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART) *t]^2}^(1/2) - { Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART) *t} *{ + Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + [Vons(PART)^2*t] } /{Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART) *t]^2}^(3/2) (endMath) /*+-----+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (time)^(-1). Rewriting in terms of Rocs(POIp(t),t) : Limit check /********************* /%>>>>>>>>> ∂[∂(t): Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))] /********************* >>>>>>>>> Figure "Calculus for a POIp", it is clear BY INSPECTION that : (mathH) ∂[∂(t): Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))] = ∂[∂(t): ROPI2ods(POIp(t),t)] = 0 (endMath) /%This makes sense, as from the Figure, Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t)) is a constant because the endpoint of Rocv is always on Line(POIo), parallel to Line(PART), the trajectory of the center of the particle. Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t)) = Rpcs(POIp)*sin(Aθpc(POIp)) /*+-----+ ALTERNATIVE DERIVATION : This also acts as a check on [consistency, correctness] of formulae. /%(2) ∂[∂(t): Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))] = ∂[∂(t): Rocs(POIp(t),t)]*sin(Aθoc(POIp(t),t)) + Rocs(POIp(t),t)*∂[∂(t): sin(Aθoc(POIp(t),t))] FIRST TERM from "∂[∂(t): Rocs(POIp(t),t)] = ∂[∂(t): |Rocv(POIp(t),t)|]" : (7) ∂[∂(t): Rocs(POIp(t),t)] = ( Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + Vons(PART)^2*t } /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + (Vons(PART) *t)^2 }^(1/2) from "Relating [Roc,sin(Aθoc(POIo)),... ] @(POIp(t),t) to [Rpcv,sin(Aθpc(POIp)),...] @(POIp)" (4) sin(Aθoc(POIp(t),t)) = Rpcs(POIp)*sin(Aθpc(POIp)) /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) So : (3) ∂[∂(t): Rocs(POIp(t),t)]*sin(Aθoc(POIp(t),t)) = ( Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + Vons(PART)^2*t } /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + (Vons(PART) *t)^2 }^(1/2) * Rpcs(POIp)*sin(Aθpc(POIp)) /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART) *t]^2 }^(1/2) = ( Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + Vons(PART)^2*t } * Rpcs(POIp)*sin(Aθpc(POIp)) /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART) *t]^2 } SECOND TERM from "∂[∂(t): sin(Aθoc(POIp(t),t))]" : (5) ∂[∂(t): sin(Aθoc(POIp(t),t))] = -Rpcs(POIp)*sin(Aθpc(POIp)) *Vons(PART) * [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / { [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 }^(3/2) /*from "Relating [Rocv,Rocs,ROPI2ocs ,rO0ocs,sin(Aθoc),cos(Aθoc)]@(POIp(t),t) to [...]@(POIp)" /% (2) Rocs(POIp(t),t)) = { Rpcs(POIp)^2 + 2*Rpcs(POIp) *cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) therefore : (4) Rocs(POIp(t),t)*∂[∂(t): sin(Aθoc(POIp(t),t))] = { Rpcs(POIp)^2 + 2*Rpcs(POIp) *cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) * -Rpcs(POIp)*sin(Aθpc(POIp)) *Vons(PART) * [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / { [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 }^(3/2) = -Rpcs(POIp)*sin(Aθpc(POIp)) *Vons(PART) * [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / { Rpcs(POIp)^2 + 2*Rpcs(POIp) *cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } PUTTING IT ALL TOGETHER : Subbing (3)&(4) into (2) : (2) ∂[∂(t): Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))] = ∂[∂(t): Rocs(POIp(t),t)]*sin(Aθoc(POIp(t),t)) + Rocs(POIp(t),t)*∂[∂(t): sin(Aθoc(POIp(t),t))] = { ( Rpcs(POIp)*cos(Aθpc(POIo(t),t)) *Vons(PART) + Vons(PART)^2*t } * Rpcs(POIp)*sin(Aθpc(POIp)) / { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t)) *Vons(PART)*t + [Vons(PART) *t]^2 } } +{ -Rpcs(POIp)*sin(Aθpc(POIp)) *Vons(PART) * [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t)) *Vons(PART)*t + [Vons(PART) *t]^2 } } = { ( Rpcs(POIp)^2*sin(Aθpc(POIp))*cos(Aθpc(POIo(t),t)) *Vons(PART) + Rpcs(POIp) *sin(Aθpc(POIp)) *Vons(PART)^2*t } / { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t)) *Vons(PART)*t + [Vons(PART) *t]^2 } } +{ -Rpcs(POIp) *sin(Aθpc(POIp)) *Vons(PART)*Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + -Rpcs(POIp) *sin(Aθpc(POIp)) *Vons(PART)*Vons(PART)*t ] / { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART) *t]^2 } } =[{ (Rpcs(POIp)^2*sin(Aθpc(POIp))*cos(Aθpc(POIo(t),t))*Vons(PART) + Rpcs(POIp) *sin(Aθpc(POIp)) *Vons(PART)^2*t } } -{ Rpcs(POIp)^2*sin(Aθpc(POIp))*cos(Aθpc(POIo(t),t))*Vons(PART) + Rpcs(POIp) *sin(Aθpc(POIp)) *Vons(PART)^2*t ] } ] /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t)) *Vons(PART)*t + [Vons(PART) *t]^2 } = 0 /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t)) *Vons(PART)*t + [Vons(PART) *t]^2 } = 0 (5) ∂[∂(t): Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))] = 0 /*As expected. /*+-----+ Limit checks : Dimensional consistency - OK in intermediate equations, as all terms reduce to (length/time). The agreement between the [geometric, algebraic] approaches above provides a small but essential degree of confirmation of the result. /********************* /%>>>>>>>>> ∂[∂(t): Rocs(POIp(t),t)*cosAθoc(POIp(t),t)] /*From the Figure "Calculus for a POIp" : /% (mathH) ∂[∂(t): Rocs(POIp(t),t)*cosAθoc(POIp(t),t)] = ∂[∂(t): Rθ0ocs(POIp(t),t)] = Vons(PART) (endMath) /*This makes sense, as from the Figure, Rocs(POIp(t),t)*cosAθoc(POIp(t),t)increases directly with Vons(PART). /*+-----+ ALTERNATIVE DERIVATION This also acts as a check on [consistency, correctness] of formulae. /%(2) ∂[∂(t): Rocs(POIp(t),t)*cosAθoc(POIp(t),t)] = ∂[∂(t): Rocs(POIp(t),t)]*cosAθoc(POIp(t),t) + Rocs(POIp(t),t)*∂[∂(t): cosAθoc(POIp(t),t)] /*+-----+ /%FIRST TERM : ∂[∂(t): Rocs(POIp(t),t)]*cosAθoc(POIp(t),t) from "∂[∂(t): Rocs(POIp(t),t)] = ∂[∂(t): |Rocv(POIp(t),t)|]" : (7) ∂[∂(t): Rocs(POIp(t),t)] = ( Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + Vons(PART)^2*t } /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + (Vons(PART) *t)^2 }^(1/2) /*from "Relating [Rocv,Rocs,ROPI2ocs ,rO0ocs,sin(Aθoc),cos(Aθoc)]@(POIp(t),t) to [...]@(POIp)" : /% (6) cos(Aθoc(POIp(t),t)) = [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] /{Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (7)&(6) into (2) : (3) ∂[∂(t): Rocs(POIp(t),t)]*cos(Aθoc(POIp(t),t)) = ( Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + Vons(PART)^2*t } /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + (Vons(PART) *t)^2 }^(1/2) *[ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) = { Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + Vons(PART)^2*t } *[ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } /*+-----+ /%SECOND TERM : Rocs(POIp(t),t)*∂[∂(t): cosAθoc(POIp(t),t)] /*from "Relating [Rocv,Rocs,ROPI2ocs ,rO0ocs,sin(Aθoc),cos(Aθoc)]@(POIp(t),t) to [...]@(POIp)" /% (2) Rocs(POIp(t),t)) = { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) from "∂[∂(t): cos(Aθoc(POIp(t),t))]" : (1) ∂[∂(t): cosAθoc(POIp(t),t)] = Vons(PART) /{Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) - [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] *{ Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + [Vons(PART)^2*t] } /{Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) therefore : (4) Rocs(POIp(t),t)*∂[∂(t): cosAθoc(POIp(t),t)] = {Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) *{ Vons(PART) /{Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) -[ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] *{ Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + [Vons(PART)^2*t] } /{Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) } = Vons(PART) -[ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] *{ Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + [Vons(PART)^2*t] } /{Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } /*+-----+ PUTTING IT ALL TOGETHER : Subbing (3)&(4) into (2) : /% (2) ∂[∂(t): Rocs(POIp(t),t)*cosAθoc(POIp(t),t)] = ∂[∂(t): Rocs(POIp(t),t)]*cosAθoc(POIp(t),t) + Rocs(POIp(t),t)*∂[∂(t): cosAθoc(POIp(t),t)] ={ ( Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + Vons(PART)^2*t ) * [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART) *t ] /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } } + { Vons(PART) - [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] *{ Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + [Vons(PART)^2*t] } /{Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } } = Vons(PART) + [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART) *t ] *( Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + Vons(PART)^2*t } /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } - [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART) *t ] *{ Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + [Vons(PART)^2*t] } /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } = Vons(PART) (5) ∂[∂(t): Rocs(POIp(t),t)*cosAθoc(POIp(t),t)] = Vons(PART) /*As expected... Limit checks : Dimensional consistency - OK in result & intermediate equations, as all terms reduce to (length/time). The agreement between the [geometric, algebraic] approaches above provides a small but essential degree of confirmation of the result. endsection /********************* >>>>>>>>> Figure "Electrostatic field basics & calculus for a POIo" http://www.BillHowell.ca/ /media/bill/SWAPPER/Lucas - Universal Force/Images/Howell - Chapter 4 - Electrostatic field basics & calculus - cropped.png Differential vector geometry of ∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)] Looking at Figure "Electrostatic field basics & calculus for a POIo" : IMPORTANT!! I must apply the spherical coordinate derivative formulation to get the same result!!!!!!!! /*+--+ /********************* >>>>>>>>> DEFINITIONS /%1a) Aθpd(RDEpdh(POIo(t),∂(t))) is the direction perpendicular to Rpch(POIo(t),t), rotated in the positive Aθpn (or Aθon) direction (i.e at an angle of PI/2 from Rpch(POIo(t),t)). This gives an angle with respect to the (RFo) theta coordinate vector, Aθpc(POIo(t),t), of (PI/2). Aθpd(RDEpdh(POIo(t),∂(t))) defines a tanjent direction, and it is used below as a reference direction for chord that is important to derive the differential it is used below as ????????? Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 1b) RDEpdh(POIo(t),∂(t)) is a unit vector (dimensionless) in the direction of Aθpd(RDEpdh(POIo(t),∂(t))) 1c) RDEpds(POIo(t),∂(t)) = |RDEpdh(POIo(t),∂(t))| = 1 is the magnitude of the unit vector RDEpdh(POIo(t),∂(t)) 1d) E0DIFF_odv(POIo(t),∂(t)) = ∂[∂(t): E0odv(POIo,t)] is the vector DIFFerential change in E0odv(POIo,t)] 1e) E0DIFF_ods(POIo(t),∂(t)) = |∂[∂(t): E0odv(POIo,t)]| = E0DIFF_ods(POIo(t),∂(t)) is the magnitude of the DIFFerential change in E0odv(POIo,t)] 1f) Aθ_E0DIFFoda(POIo(t),∂(t)) is the direction of ∂[∂(t): E0odv(POIo,t)] with respect to E0odv(POIo,t), the latter being the same direction as Rpcv(POIo(t),t) 1g) Aθ_E0DIFFoca(POIo(t),∂(t)) is the direction of ∂[∂(t): E0odv(POIo,t)] with respect to E0odv(POIo,t), the latter being the same direction as Rpcv(POIo(t),t) : Aθ_E0DIFFoca(POIo(t),∂(t)) = Aθ_E0DIFFoca(POIo(t),∂(t)) + Aθpd(RDEpdh(POIo(t),∂(t))) + Aθpc(POIo(t),t) = Aθ_E0DIFFoca(POIo(t),∂(t)) + PI/2 + Aθpc(POIo(t),t) Looking at Figure "∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)]" : (1) ∂[∂(t): E0pdv(POIo(t),t)] = E0pds(POIo(t),t)*∂[∂(t): Aθpc(POIo(t),t)]*RDEpdh(POIo(t),∂(t)) + ∂[∂(t): |E0pdv(POIo(t),t)|] *Rpch(POIo(t),t) /*... ignoring differential changes in directions /%(2) ∂[∂(t): E0odv(POIo,t)] = E0ods(POIo,t)*∂[∂(t): Aθpc(POIo(t),t)]*RDEpdh(POIo(t),∂(t)) + ∂[∂(t): |E0odv(POIo,t)|] *Rpch(POIo(t),t) /*... ignoring differential changes in directions /********************* /%>>>>>>>>> ∂[∂(t): E0odv(POIo)] ≠ ∂[∂(t): E0pdv(POIp)] = 0 /*Reminders for Chapter 4 : Vonv(PART) = constant [Particle, observer] frames of reference (RFp) & (RFo) are IDENTICAL [scale, rotation, etc] at time t=0, apart from the given that the particle's reference frame (RFp) moves with velocity Vonv(PART) relative to RFo. Reminders for (POIo) - Point Of Interest that is FIXED in the observer reference frame (RFo) : [Roc,Ooc,Poc] and their derived concepts, are NOT functions of time, i.e. (POIp). Their derivatives with respect to time are zero. [Rpc,Opc,Ppc], their derived concepts, and their derivatives are all functions of time, i.e. (POIo(t),t) /*+-----+ (RFp) basis /%Looking at Figure "∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)]" : From "Differential geometry of ∂[∂(t): BTodv(POIo,t)] = ∂[∂(t): BTpdv(POIo(t),t)]" : ∂[∂(t): E0pdv(POIo(t),t)] = E0pds(POIo(t),t)*∂[∂(t): Aθpc(POIo(t),t)] *RDEpdh(POIo(t),∂(t)) + ∂[∂(t): E0pds(POIo(t),t)]*Rpch(POIo(t),t) ... ignoring differential change in direction for RDEpdh(POIo(t),∂(t)) & Rpch(POIo(t),t) /*+-----+ FIRST TERM /%(1) E0pds(POIo(t),t)*∂[∂(t): Aθpc(POIo(t),t)]*RDEpdh(POIo(t),∂(t)) where, as in "Differential vector geometry of ∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)]" : 1a) Aθpd(RDEpdh(POIo(t),∂(t))) is the direction perpendicular to Rpch(POIo(t),t), rotated in the positive Aθpn (or Aθon) direction (i.e at an angle of PI/2 from Rpch(POIo(t),t)). This gives an angle with respect to the (RFo) theta coordinate vector of (PI/2). Aθpd(RDEpdh(POIo(t),∂(t))) defines a tanjent direction, and it is used below as a reference direction for chord that is important to derive the differential it is used below as ????????? 1c) RDEpdh(POIo(t),∂(t)) is a unit vector in the direction of Aθpd(RDEpdh(POIo(t),∂(t))) From "E0ods(POIo,t) = E0pds(POIo(t),t)" : (1) E0pds(POIo(t),t) = Q(PART)/Rpcs(POIp)^2 From "∂[∂(t): Aθpc(POIo(t),t)]" : (5) d[∂(t): Aθpc(POIo(t),t)] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) Subbing (1)* & (5)* into (1) : (2) E0pds(POIo(t),t)*∂[∂(t): Aθpc(POIo(t),t)] *RDEpdh(POIo(t),∂(t)) = Q(PART) /Rpcs(POIp)^2 *Vons(PART)/Rpcs(POIo(t),t) *sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) = Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3*sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) /*+--+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (charge/length^2/time). (ignoring electric permeability for Gaussian coordinates) /*+-----+ SECOND TERM /%(3) ∂[∂(t): E0pds(POIo(t),t)]*Rpch(POIo(t),t) ... ignoring differential change in direction From "∂[∂(t): E0pds(POIo(t),t)] = ∂[∂(t): E0pds(POIo(t),t)]" : 4*) ∂[∂(t): E0pds(POIo(t),t)] = 2*Q(PART)*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIp)^3 Subbing (4)* into (3) : (4) ∂[∂(t): |E0pdv(POIo(t),t)|] = 2*Q(PART)*Vons(PART)/Rpcs(POIp)^3*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) /*+--+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (charge/length^2/time). (ignoring electric permeability for Gaussian coordinates) /*+-----+ COMBINING TERMS Subbing (2)&(4) into (1) : /% (1) ∂[∂(t): E0pdv(POIo(t),t)] = E0pds(POIo(t),t)*∂[∂(t): Aθpc(POIo(t),t)] *RDEpdh(POIo(t),∂(t)) + ∂[∂(t): |E0pdv(POIo(t),t)|] *Rpch(POIo(t),t) = Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3*sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*Q(PART)*Vons(PART)/Rpcs(POIp)^3*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) /*Therefore : /% ∂[∂(t): E0pdv(POIo(t),t)] = Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3*sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*Q(PART)*Vons(PART)/Rpcs(POIp)^3*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) /*+--+ /%Looking at Rpch(POIo(t),t), RDEpdh(POIo(t),∂(t)) By definition, Rpch(POIo(t),t) is in the direction of Rpcv(POIo(t),t), at angle Aθpc(POIo(t),t). /*From "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)" : /% (1) Rpcv(POIo(t),t) = Rocv(POIo) - Vonv(PART)*t From "Differential vector geometry of ∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)]" : 1a) Aθpd(RDEpdh(POIo(t),∂(t))) is the direction perpendicular to Rpch(POIo(t),t), rotated in the positive Aθpn (or Aθon) direction (i.e at an angle of PI/2 from Rpch(POIo(t),t)). This gives an angle with respect to the (RFo) theta coordinate vector, Aθpc(POIo(t),t), of (PI/2). Aθpd(RDEpdh(POIo(t),∂(t))) defines a tanjent direction, and it is used below as a reference direction for chord that is important to derive the differential it is used below as ????????? Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 1b) RDEpdh(POIo(t),∂(t)) is a unit vector (dimensionless) in the direction of Aθpd(RDEpdh(POIo(t),∂(t))) 1c) RDEpds(POIo(t),∂(t)) = |RDEpdh(POIo(t),∂(t))| = 1 is the magnitude of the unit vector RDEpdh(POIo(t),∂(t)) Summary : (mathH) ∂[∂(t): E0pdv(POIo(t),t)] = Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t)] (endMath) where : Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 /*+-----+ (RFo) basis : Where convenient, express variables as functions of FIXED variables : (POIp) -> -Vodv(PART),t,Rpcv(POIp),Aθpc(POIp),Aφpc(POIp) (POIo) -> Vodv(PART),t,Rocv(POIo),Aθoc(POIo),Aφoc(POIo) From "Relating [Rpcs,RO0pcs,ROPI2pcs,sin(Aθpc),cos(Aθpc)]@t to [Roc,AOo,APo] for (POIo)" : /% (2) Rpcs(POIo(t),t)) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) (5) sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) (6) cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) Subbing (2)*,(5)*(6)* into (6) : (6) ∂[∂(t): E0pdv(POIo(t),t)] = Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[ sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) ] where : Aθpd(Rpch(POIo(t),t)) = Aθpc(POIo(t),t) Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 = Q(PART)*Vons(PART) *[ RDEpdh(POIo(t),∂(t)) * Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) + Rpch(POIo(t),t) *2*[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^3 = Q(PART)*Vons(PART) *[ RDEpdh(POIo(t),∂(t)) * Rocs(POIo)*sin(Aθoc(POIo)) + Rpch(POIo(t),t) *2*[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^3 / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) = Q(PART)*Vons(PART) *[ RDEpdh(POIo(t),∂(t)) * Rocs(POIo)*sin(Aθoc(POIo)) + Rpch(POIo(t),t) *2*[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 (7) = Q(PART)*Vons(PART) *[ Rocs(POIo)*sin(Aθoc(POIo)) *RDEpdh(POIo(t),∂(t)) + 2*[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rpch(POIo(t),t) ] ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 /*+--+ Conversions of directions : From (6) : /% Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 /*As above, from "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)" : /% (5) sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) (6) cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) So Aθpc(POIo(t),t) can be expressed either as : (8) Aθpc(POIo(t),t) = arcsin [ Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ] or = arccos [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) ] Summarizing (7),(8),(9) : (mathH) ∂[∂(t): E0pdv(POIo(t),t)] = Q(PART)*Vons(PART) * {Rocs(POIo)*sin(Aθoc(POIo))*RDEpdh(POIo(t),∂(t)) + 2*[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rpch(POIo(t),t)] } / {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 (endMath) where : Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 Aθpc(POIo(t),t) = arcsin [ Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ] = arccos [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) ] /*+--+ LIMIT CHECKS : Dimensional consistency for magnitude - OK, as all units reduce to (charge/length^2/t) Dimensional consistency for direction - OK, as all terms reduce to (radians). ... actually - issue of dimensionless versus radians??? /********************* /%>>>>>>>>> ∂[∂(t): E0ods(POIo,t)], using proper E0odv(POIo,t) vector approach /*see Figure "Electrostatic field basics & calculus for a POIo", starting with knowledge that the electrostatic field at a point is the same for RFp as for RFo, therefore : /% ∂[∂(t): E0ods(POIo,t)] = ∂[∂(t): E0pds(POIo(t),t)] and : (mathH) ∂[∂(t): E0pds(POIo(t),t)] = ∂[∂(t): |E0pdv(POIo(t),t)|] (endMath) /*+-----+ (RFp) FORMAT /%From "E0odv(POIo,t) = E0pdv(POIo(t),t)" : 1*) E0pdv(POIo(t),t) = E0odv(POIo,t) = Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) Subbing (1)* into (1) : (1) ∂[∂(t): E0pds(POIo(t),t)] = ∂[∂(t): |E0pdv(POIo(t),t)|] = ∂[∂(t): |Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t)|] (2) = |Q(PART)| *dot* ∂[∂(t): |Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2|] /*From "Howell's use of Kahan vector derivative formulation" : (1)** ∂[∂(t): |z|] = z_T *dot* ∂[∂(t):z] / |z| where *dot* = dotProduct where : z, ∂[∂(t):z] are column vectors, z_T is a row vector _T indicates [vector, matrix] transpose * is matrix multiplication (here z_T*∂[∂(t):z] is same as dotProduct as yields 1row*1column result |...| denotes norm question: can one assume that "z_T" is the linear functional dual to z wrt |z|? Applying the template (1)** to the derivative of (2) : /%(3) ∂[∂(t): |Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2|] = [ Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2 ]_T *dot* ∂[∂(t): Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2] / | Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2 | /*+--+ Now looking at : /%(4) ∂[∂(t): Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2] = ∂[∂(t): Rpch(POIo(t),t)]/Rpcs(POIo(t),t)^2 + Rpch(POIo(t),t)*∂[∂(t): Rpcs(POIo(t),t)^( - 2)] First derivative term from "∂[∂(t): Rpch(POIo(t),t)]" Equation (2) : (2)* ∂[∂(t): Rpch(POIp)] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*RDEpdh(POIo(t),t) where RDEpdh(POIo(t),t) is anchored at end of Rpch(POIo(t),t) and is at angle Aθpc(POIo(t),t) + PI/2, ie perpendicular to Rpch(POIo(t),t) angle Aφpc(POIo(t),t) doesn't change Second term's derivative : (5) ∂[∂(t): Rpcs(POIo(t),t)^( - 2)] = -2*Rpcs(POIo(t),t)^(-3)*∂[∂(t): Rpcs(POIo(t),t)] Subbing (2)* & (5) into (4) : (4) ∂[∂(t): Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2] = ∂[∂(t): Rpch(POIo(t),t)]/Rpcs(POIo(t),t)^2 + Rpch(POIo(t),t)*∂[∂(t): Rpcs(POIo(t),t)^( - 2)] = ∂[∂(t): Rpch(POIo(t),t)] /Rpcs(POIo(t),t)^2 + Rpch(POIo(t),t) *∂[∂(t): Rpcs(POIo(t),t)^( - 2)] (6) = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*RDEpdh(POIo(t),t) /Rpcs(POIo(t),t)^2 + Rpch(POIo(t),t) *-2*Rpcs(POIo(t),t)^(-3) *∂[∂(t): Rpcs(POIo(t),t)] From "∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|]" : (1)* ∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|] = -Vons(PART)*cos(Aθpc(POIo(t),t)) Subbing (1)* into (6) : ∂[∂(t): Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*RDEpdh(POIo(t),t) /Rpcs(POIo(t),t)^2 + Rpch(POIo(t),t) *-2*Rpcs(POIo(t),t)^(-3) *-Vons(PART)*cos(Aθpc(POIo(t),t)) (7) = [ Vons(PART)*sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),t) + 2*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) ]/Rpcs(POIo(t),t)^3 /*+--+ Subbing (7) into (3) : (3) ∂[∂(t): |Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2| ] = [ Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2 ]_T *dot* ∂[∂(t): Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2 ] / | Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2 | Keeping the order listed of vectors : = [ Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2 ]_T *dot* [ Vons(PART)*sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),t) + 2*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) ] / Rpcs(POIo(t),t)^3 / | Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2 | = Rpch(POIo(t),t)_T / Rpcs(POIo(t),t)^2 * Vons(PART) *dot* [ sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),t) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) ] / Rpcs(POIo(t),t)^3 / | Rpch(POIo(t),t) | * Rpcs(POIo(t),t)^2 =* [ sin(Aθpc(POIo(t),t))*Rpch(POIo(t),t)_T *dot* RDEpdh(POIo(t),t) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t)_T *dot* Rpch(POIo(t),t) ] * Vons(PART) / | Rpch(POIo(t),t) | / Rpcs(POIo(t),t)^2 / Rpcs(POIo(t),t)^3 * Rpcs(POIo(t),t)^2 (8) =* [ sin(Aθpc(POIo(t),t))*Rpch(POIo(t),t)_T *dot* RDEpdh(POIo(t),t) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t)_T *dot* Rpch(POIo(t),t) ] * Vons(PART) / | Rpch(POIo(t),t) | / Rpcs(POIo(t),t)^3 But : (9) | Rpch(POIo(t),t) | = 1, from "∂[∂(t): Rpch(POIo(t),t)]" (see above) : RDEpdh(POIo(t),t) is anchored at end of Rpch(POIo(t),t) and is at angle Aθpc(POIo(t),t) + PI/2, ie perpendicular to Rpch(POIo(t),t) Rpch(POIo(t),t)_T *dot* RDEpdh(POIo(t),t), where *dot* => dotProduct (see Kreyszig p200) BUT - is v_T perpendicular to v???? ?NO? - no effect here? = |Rpch(POIo(t),t)| * |RDEpdh(POIo(t),t)|*cos(Aθpc(POIo(t),t) + PI/2 - Aθpc(POIo(t),t)) = |Rpch(POIo(t),t)| * |RDEpdh(POIo(t),t)|*cos(PI/2) = |Rpch(POIo(t),t)| * |RDEpdh(POIo(t),t)|*0 (10) = 0 For : Rpch(POIo(t),t)_T *dot* Rpch(POIo(t),t) = |Rpch(POIo(t),t)| * |Rpch(POIo(t),t)|*cos(0) = 1 * 1 * 1 (11) = 1 Subbing (9),(10),(11) into (8) : ∂[∂(t): |Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2| ] =* [ sin(Aθpc(POIo(t),t))*Rpch(POIo(t),t)_T*RDEpdh(POIo(t),t) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t)_T*Rpch(POIo(t),t) ] * Vons(PART) / | Rpch(POIo(t),t) | / Rpcs(POIo(t),t)^3 =* [ sin(Aθpc(POIo(t),t))*0 + 2*cos(Aθpc(POIo(t),t))*1 ] * Vons(PART) / 1 / Rpcs(POIo(t),t)^3 = 2*cos(Aθpc(POIo(t),t))*Vons(PART)/Rpcs(POIo(t),t)^3 /*Key intermediate result useful elsewhere : /%(12) ∂[∂(t): |Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2|] = 2*cos(Aθpc(POIo(t),t))*Vons(PART)/Rpcs(POIo(t),t)^3 /*Subbing (12) into (2) : /% (2) ∂[∂(t): E0pds(POIo(t),t),t] = |Q(PART)|*∂[∂(t): |Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2|] = |Q(PART)|*2*cos(Aθpc(POIo(t),t))*Vons(PART)/Rpcs(POIo(t),t)^3 = 2*|Q(PART)|*cos(Aθpc(POIo(t),t))*Vons(PART)/Rpcs(POIo(t),t)^3 /*Summarizing : /% (mathH) ∂[∂(t): E0ods(POIo,t)] = ∂[∂(t): E0pds(POIo(t),t)] = 2*|Q(PART)|*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 (endMath) Note : This is the SAME as the Rpch component of " ∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)]" : 6*) ∂[∂(t): E0pdv(POIo(t),t)] = Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[ sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) ] where : Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 This should be expected, as per the Figure "∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)]". /*-----+ LIMIT CHECKS /*--+ Does this make sense? /*POIo on line of trajectory of particle as a limit case : Over delta_time of dt, E changes by : initial : Ei = Q / R(t)^2 end : Et = Q / R(t+dt)^2 where R(t+dt) = R(t) + V*dt so dE = Et - Ei = Q*(1/R(t+dt)^2 - 1/R(t)^2) = Q*(1/[R(t) + V*dt]^2 - 1/R(t)^2) = Q*{ R(t)^2 - [R(t) + V*dt]^2 } / { [R(t) + V*dt]^2 * R(t)^2) } = Q*{ R(t)^2 - R(t)^2 - 2*R(t)*V*dt - (V*dt)^2 } / { [R(t)^2 + 2*R(t) *V*dt + (V*dt)^2] * R(t)^2)] = Q*{ - 2*R(t)*V*dt - (V*dt)^2 } / { R(t)^4 + 2*R(t)^3*V*dt + (V*dt)^2 * R(t)^2 } Trick, for very small dt, ignore squared terms " = Q*{ - 2*R(t)*V*dt } / { R(t)^4 + 2*R(t)^3*V*dt } = -2*Q*R(t)*V*dt / R(t)^2 / [ R(t)^2 + 2*R(t)*V*dt ] Also, denominator - ignore dt term = -2*Q*R(t)*V*dt / R(t)^2 / R(t)^2 = -2*Q*R(t)*V*dt / R(t)^4 = -2*Q *V*dt / R(t)^3 So 13a) d[dt : E] = -2*Q*V/R(t)^3 This is the same as (13) except for minus sign, and cosine term Sign - when R < 0, cos < 0, so (13) is +ve, and (13a) is +ve when R > 0, cos > 0, so (13) is -ve, and (13a) is -ve OK!! /*--+ Second check, take tp when POIp is perpendicular to particle : dE = Et - Ei = Q*(1/R(t+dt)^2 - 1/R(t)^2), for absolute values of Q, R = Q*(1/{ [R(t)^2 + (V*dt)^2]^(1/2) }^2 - 1/R(t)^2) = Q*(1/{ [R(t)^2 + (V*dt)^2] } - 1/R(t)^2) = Q*(1/{ [R(t)^2 + (V*dt)^2] } - 1/R(t)^2) /*--+ Dimensional check /% (13) ∂[∂(t): E0ods(POIo,t)] = ∂[∂(t): E0pds(POIo(t),t)] = 2*|Q(PART)|*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 = charge * length/time / length^3 = charge / time / length^2 /*From "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt" : dimensions of E = charge/length^2 therefore d[dt : E] units are charge/length^2/time This is OK /*+-----+ (RFo) FORMAT From "Rpcs(POIo(t),t)" : /% (3)* Rpcs(POIo(t),t) = |Rpcv(POIo(t),t)| = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) From "cos(Aθpc(POIo(t),t))" : (1)* cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) Subbing (3)*&(1)* into (13) : (13) ∂[∂(t): E0pds(POIo(t),t)] = 2*|Q(PART)|*cos(Aθpc(POIo(t),t))*Vons(PART)/Rpcs(POIo(t),t)^3 = 2*|Q(PART)|*Vons(PART) *cos(Aθpc(POIo(t),t)) /Rpcs(POIo(t),t)^3 = 2*|Q(PART)|*Vons(PART) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^3 = 2*|Q(PART)|*Vons(PART) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^2 Summarizing : (mathH) ∂[∂(t): E0ods(POIo,t)] = ∂[∂(t): E0pds(POIo(t),t)] = 2*|Q(PART)|*Vons(PART) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^2 (endMath) /*+--+ LIMIT CHECKS : Dimensional consistency for magnitude - OK, as all units reduce to (charge/length^2/t) Dimensional consistency for direction - OK, as all terms reduce to (radians). ... actually - issue of dimensionless versus radians??? /********************* >>>>>>>>> ∂[∂(t): E0ods(POIo,t)] = ∂[∂(t): E0pds(POIo(t),t)] - cheating E0ods(POIo,t) scalar approach WARNING : Should do vector differentiation THEN take the magnitude!?? -> this was done above (after the cheating approach) Given the parametric equation for E0ds(POIo(t),t), perhaps this isn't a problem? Actually, it WORKS! see Figure "∂[∂(t): E0odv(POIo,t)] - basic metrics" /%From "E0ods(POIo,t) = E0pds(POIo(t),t)" : (2) E0ods(POIo,t) = |Q(PART)|/Rpcs(POIp(t),t)^2 and Rpch is a unit vector, so : (1) |Rpch(POIp(t),t)| = 1 So : (2) ∂[∂(t): E0ods(POIo,t)] = ∂[∂(t): |Q(PART)|/Rpcs(POIp(t),t)^2] = |Q(PART)|/Rpcs(POIp(t),t)*∂[∂(t): Rpcs(POIp(t),t)^( - 3)] From "∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)]" : (1) ∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|] = -Vons(PART)*cos(Aθpc(POIo(t),t)) Subbing (1)* into (2) : (3) ∂[∂(t): E0pds(POIo(t),t)] = ∂[∂(t): |Q(PART)|/Rpcs(POIp(t),t)^2] = |Q(PART)|*(-2)/Rpcs(POIp(t),t)^3*-Vons(PART)*cos(Aθpc(POIo(t),t)) /*For now - ARBITRARY sign on equations for scalar derivatives, as I have to go back & fix the derivations (not as straightforward as it sounds!). /%∂[∂(t): EIods(POIo(t),t)] has the opposite sign of ∂[∂(t): E0ods(POIo,t)] Note that for : /% -PI/2 <= Aθpc(POIo(t),t)) <= PI/2 : then cos(Aθpc(POIo(t),t)) >0 and ∂[∂(t): E0pds(POIo(t),t)] < 0, and ∂[∂(t): EIpds(POIo(t),t)] < 0 /*19Dec2017 The following is nonsensical, as PI/2 >= -PI/2 !!?? /% PI/2 <= Aθpc(POIo(t),t)) <= -PI/2 : then cos(Aθpc(POIo(t),t)) <0 and ∂[∂(t): E0pds(POIo(t),t)] > 0, and ∂[∂(t): EIpds(POIo(t),t)] > 0 /*For Vonv(PART) > 0 (as per definition of BOTH coordinate frames of reference!), these results make sense Finally : /% (mathH) ∂[∂(t): E0pds(POIo(t),t)] = 2*|Q(PART)|*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIp(t),t)^3 (endMath) /*Actually, this cheating approach WORKS! /*+-----+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (charge/length^2/time). (ignoring electric permeability for Gaussian coordinates) Check scalar versus vector approaches : /%From "∂[∂(t): E0ods(POIo,t)] = ∂[∂(t): E0pds(POIo(t),t)] - proper E0odv(POIo,t) vector approach" : (3) ∂[∂(t): E0pds(POIo(t),t)] = |Q(PART)|*Vons(PART)/Rpcs(POIo(t),t)^3*[1 + 3*cos(Aθpc(POIo(t),t))^2] /*This is interesting as the difference between the vector and scalar(incorrect) approaches is in the constant coefficients and cos terms : /%Vector : [1 + 3*cos(Aθpc(POIo(t),t))^2] which is in the interval [1,4] /*Scalar : 2 /*+-----+ Reminders for Chapter 4 : OBJECTIVES : Where convenient, express variables as functions of FIXED variables : (POIp) -> -Vodv(PART),t,Rpcv(POIp),Aθpc(POIp),Aφpc(POIp) (POIo) -> Vodv(PART),t,Rocv(POIo),Aθoc(POIo),Aφoc(POIo) /*From "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)" : /% (2) Rpcs(POIo(t),t)) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /*From "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)" : /% (6) cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (2)* and (6)* into (4) : (4) ∂[∂(t): E0ods(POIo,t)] = 2*|Q(PART)|*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIp)^3 = 2*|Q(PART)|*Vons(PART) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^3 = 2*|Q(PART)|*Vons(PART) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 /*For now - ARBITRARY sign on equations for scalar derivatives, as I have to go back & fix the deerivations (not as straightforward as it sounds!). ∂[∂(t): EIods(POIo(t),t)] has the opposite sign of ∂[∂(t): E0ods(POIo,t)] Note that for : /% -PI/2 <= Aθpc(POIo(t),t)) <= PI/2 : then cos(Aθpc(POIo(t),t)) >0 and ∂[∂(t): E0pds(POIo(t),t)] < 0, and ∂[∂(t): EIpds(POIo(t),t)] < 0 /*19Dec2017 The following is nonsensical, as PI/2 >= -PI/2 !!?? /% PI/2 <= Aθpc(POIo(t),t)) <= -PI/2 : then cos(Aθpc(POIo(t),t)) <0 and ∂[∂(t): E0pds(POIo(t),t)] > 0, and ∂[∂(t): EIpds(POIo(t),t)] > 0 /*For Vonv(PART) > 0 (as per definition of BOTH coordinate frames of reference!), these results make sense Summarizing : /% (mathH) ∂[∂(t): E0ods(POIo,t)] = 2*|Q(PART)|*Vons(PART) *{ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 (endMath) /*Actually, it WORKS! /*+--+ LIMIT CHECKS : see "proper vector approach" above these derivations /********************* >>>>>>>>> Figure "∂[∂(t): BTodv(POIo,t)]" http://www.BillHowell.ca/ /media/bill/SWAPPER/Lucas - Universal Force/Images/Howell - Chapter 4 - dp[dt _ BTodv(POIo,t)] - cropped.png /********************* /%>>>>>>>>> ∂[∂(t): BTodv(POIp(t),t)] = ∂[∂(t): BTpdv(POIo(t),t)] ≠ ∂[∂(t): BTpdv(POIp)] = 0, without use of Lenz's Induction Law (need to RE-CHECK!!!) /*<<< 29Mar2018 Is it correct that ∂[∂(t): BTpdv(POIo(t),t)] ≠ 0 ? >>> <<< 30Mar2018 Must I do spherical coordinate derivatives, as did Lucas? >>> In order to compare to Lucas's intermediate results, it is handy to have an expression BEFORE invoking of Lenz's Law. Looking at Figure "∂[∂(t): BTodv(POIo,t)] = ∂[∂(t): BTpdv(POIo(t),t)]" : From "Differential geometry of ∂[∂(t): BTodv(POIo,t)] = ∂[∂(t): BTpdv(POIo(t),t)]" : /% (mathH) ∂[∂(t): E0pdv(POIo(t),t)] = E0pds(POIo(t),t)*∂[∂(t): Aθpc(POIo(t),t)]*RDEpdh(POIo(t),∂(t)) + ∂[∂(t): |E0pdv(POIo(t),t)|] *Rpch(POIo(t),t) (endMath) /*... ignoring differential change in direction for RDEpdh(POIo(t),dt) & Rpch(POIo(t),t) /*+-----+ (RFp) basis <<< 30Mar2018 Right now this does simple planar derivative (Cartesian), not spherical coordinates as Lucas did >>> /%From "BTodv(POIo,t) = BTpdv(POIo(t),t), without use of Lenz's Law" : (6) BTpdv(POIo(t),t) = BTodv(POIo,t) = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) * [ Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t) ] where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Therefore, taking the derivative of (6)* : ∂[∂(t): BTpdv(POIo(t),t)] = ∂[∂(t): Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t))*[Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t)]] = ∂[∂(t): Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t))]* [ Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t)] + Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *∂[∂(t): Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t)] (1) = ∂[∂(t): Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t))]* [ Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t)] + Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t))*{∂[∂(t): Q(PART)/Rpcs(POIo(t),t)^2]- ∂[∂(t): EIpds(POIo(t),t)] } /*+--+ /%Looking at the term in (1) ∂[∂(t): Rodh(Vonv_X_Rpcv(POIo(t),t))] From Figure "∂[∂(t): Rodh(Vonv_X_Rpcv(POIo(t),t))]" /*As with most of the models for Chapter 4, the phi (P) line-of-direction of the magnetic field B and electric field E of a [particle, POIo, POIp] does not change with time. Additionally, while the the theta (O) direction does change with time for E for a POIo (but NOT, or course, for the [particle, POIp] as [E,B] = constant = 0 for those situations), it does NOT change for B, as the latter is perpendicular to the [Vonv, Rpcv] plane, which itself is of constant P (phi) direction. Therefore, using /% Aφoc(Vonv&Rpcv(POIo(t),t)) = Angle between the plane of Vonv_X_Rpcv(POIo(t),t), and Aφod(BTodv(POIo,t)): /% Aφoc(planeofVonv&Rpcv(POIo(t),t)) = constant = Aφoc(POIo) AngleBetween(planeofVonv&Rpcv(POIo(t),t),APod(BTodv(POIo,t))) = constant = PI/2 in phi (i.e. Aφpd(Vonv_X_Rpcv(POIo(t),t))) Aφoc(BT(POIo(t),t)) = constant = Aφoc(POIo) + PI/2 (again assuming BI,BT are collinear) Neither the line-of-direction nor magnitude (obviously as this is a unit vector) of Rodh(Vonv_X_Rpcv(POIo(t),t)) change. However, while the LINE-OF-DIRECTION of Rodh(Vonv_X_Rpcv(POIo(t),t)) does not change, its sign CAN change depending on sin(Vonv_X_Rpcv(POIo(t),t)), which is just sin(Aθpc(POIo(t),t))!! Therefore : (2) ∂[∂(t): Rodh(Vonv_X_Rpcv(POIo(t),t))] = sign[sin(Vonv_X_Rpcv(POIo(t),t))] = [ - 1,1] Denote : (3) sgn(BV) = ∂[∂(t): Rodh(Vonv_X_Rpcv(POIo(t),t))] = sign[sin(Vonv_X_Rpcv(POIo(t),t))] = [ - 1,1] /*WRONG!!! - should do properly... Therefore this is a "kind-of-constant" and it can be moved outside from the term /%∂[∂(t): sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t))]. /*+--+ /%Bringing constants [ Q(PART),Vons(PART)/c, Rodh(Vonv_X_Rpcv(POIo(t),t))] out of the derivatives of (1), but use sgn(BV)*Rodh(Vonv_X_Rpcv(POIo(t),t)) when *Rodh(Vonv_X_Rpcv(POIo(t),t)) taken out sign(: (1) ∂[∂(t): BTpdv(POIo(t),t)] = ∂[∂(t): Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t))]*[ Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t)] + Vons(PART)/c*sin(Aθpc(POIo(t),t)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ ∂[∂(t): Q(PART)/Rpcs(POIo(t),t)^2] - ∂[∂(t): EIpds(POIo(t),t)] } = Vons(PART)/c*∂[∂(t): sin(Aθpc(POIo(t),t))]*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t)] + Vons(PART)/c *sin(Aθpc(POIo(t),t)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) *{Q(PART)*∂[∂(t): 1/Rpcs(POIo(t),t)^2] - ∂[∂(t): EIpds(POIo(t),t)] } (3) = Vons(PART)/c *Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ ∂[∂(t): sin(Aθpc(POIo(t),t))] *[Q(PART) /Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t) ] + sin(Aθpc(POIo(t),t)) *[Q(PART)*∂[∂(t): 1/Rpcs(POIo(t),t)^2]- ∂[∂(t): EIpds(POIo(t),t)] ] } /*+--+ Looking at : (4) ∂[∂(t): 1/Rpcs(POIo(t),t)^2 ] = ∂[∂(t): Rpcs(POIo(t),t)^(-2) ] = -2*Rpcs(POIo(t),t)^(-3)*∂[∂(t): Rpcs(POIo(t),t)] From "∂[∂(t): Rpcs(POIo(t),t) ] = ∂[∂(t): |Rpcv(POIo(t),t)|]" : (1) ∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)| ] = -Vons(PART)*cos(Aθpc(POIo(t),t)) Subbing (1)* into (4) : (4) ∂[∂(t): 1/Rpcs(POIo(t),t)^2 ] = ∂[∂(t): Rpcs(POIo(t),t)^(-2) ] = -2*Rpcs(POIo(t),t)^(-3)*∂[∂(t): Rpcs(POIo(t),t)] = -2*Rpcs(POIo(t),t)^(-3)*-Vons(PART)*cos(Aθpc(POIo(t),t)) (5) = 2*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 From "∂[∂(t): sin(Aθpc(POIo(t),t)) ]" : (2)* ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) /*+--+ Subbing (5) & (2)* into (3) : (3) ∂[∂(t): BTpdv(POIo(t),t)] = ∂[∂(t): BTodv(POIo,t)] = Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo)) *{ ∂[∂(t): sin(Aθpc(POIo(t),t))]*[ Q(PART) /Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t) ] + sin(Aθpc(POIo(t),t)) *[ Q(PART) *∂[∂(t): 1/Rpcs(POIo(t),t)^2]- ∂[∂(t): EIpds(POIo(t),t)] ] } = Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo)) *{ Vons(PART) *sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) *[ Q(PART) /Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t)] + sin(Aθpc(POIo(t),t))*[ Q(PART)*2*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 - ∂[∂(t): EIpds(POIo(t),t)] ] } = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo)) *{ Q(PART)*Vons(PART) *sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 - Vons(PART) *sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) *EIpds(POIo(t),t) + 2*Q(PART)*Vons(PART) *sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 - sin(Aθpc(POIo(t),t)) *∂[∂(t): EIpds(POIo(t),t)] ] } = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo)) *Vons(PART) *sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t)) *{ Q(PART)/Rpcs(POIo(t),t)^3 - /Rpcs(POIo(t),t) *EIpds(POIo(t),t) + 2*Q(PART)/Rpcs(POIo(t),t)^3 } - Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo))*sin(Aθpc(POIo(t),t))*∂[∂(t): EIpds(POIo(t),t)] = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo)) *{ 3*Q(PART)/Rpcs(POIo(t),t)^3 - EIpds(POIo(t),t)/Rpcs(POIo(t),t) } - Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo))*sin(Aθpc(POIo(t),t))*∂[∂(t): EIpds(POIo(t),t)] ={ 3*Q(PART)/Rpcs(POIo(t),t)^3*Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t)) - Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t)) *EIpds(POIo(t),t)/Rpcs(POIo(t),t) - Vons(PART) /c*sin(Aθpc(POIo(t),t))^2 *∂[∂(t): EIpds(POIo(t),t)] } *Rodh(Vonv_X_Rpcv(POIo)) ={ 3*Q(PART)*Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 - Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) *EIpds(POIo(t),t) - Vons(PART) /c*sin(Aθpc(POIo(t),t))^2 *∂[∂(t): EIpds(POIo(t),t)] } *Rodh(Vonv_X_Rpcv(POIo)) = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo)) *{ 3*Q(PART) /Rpcs(POIo(t),t)^3 - EIpds(POIo(t),t)/Rpcs(POIo(t),t) - ∂[∂(t): EIpds(POIo(t),t)]/Vons(PART)/cos(Aθpc(POIo(t),t)) } Summarising : (mathH) ∂[∂(t): BTpdv(POIo(t),t)] = ∂[∂(t): BTodv(POIo,t)] = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo)) *{3*Q(PART)/Rpcs(POIo(t),t)^3 - EIpds(POIo(t),t)/Rpcs(POIo(t),t) - ∂[∂(t): EIpds(POIo(t),t)]/Vons(PART)/cos(Aθpc(POIo(t),t))} (endMath) <<<< Improper derivative of Rodh(Vonv_X_Rpcv(POIo)) ]but use sgn(BV)*Rodh(Vonv_X_Rpcv(POIo)) when *Rodh(Vonv_X_Rpcv(POIo)) taken out /*At this stage (pre [Lenz's Law & Thomas Barnes iterations]), terms with EIpds(POIo(t),t) stay the same. /*+--+ Limit checks : 29Mar2018 Dimensional check Subbing in units of measure : /%(6)* ∂[∂(t): BTpdv(POIo(t),t)] = ∂[∂(t): BTodv(POIo,t)] (charge/length/time)/time = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) (length/time )^2 *{ 3*Q(PART) /Rpcs(POIo(t),t)^3 charge / length^3 - EIpds(POIo(t),t)/Rpcs(POIo(t),t) (charge/length^2) / length - ∂[∂(t): EIpds(POIo(t),t)]/Vons(PART)/cos(Aθpc(POIo(t),t)) (charge/length^2/time) /(length/time) } = (length/time )^2 *{ charge/length^3 - (charge/length^2) / length - (charge/length^2/time) /(length/time) } = (length/time )^2 *{ charge/length^3 - (charge/length^3) - (charge/length^3) } = (length/time )^2 * charge/length^3 = (charge/length/time^2) /*Therefore (charge/length/time^2) = (charge/length/time^2) = ∂[∂(t): B] units So the dimensions are consistent. /*+-----+ (RFo) basis /%From "BTodv(POIo,t) = BTpdv(POIo(t),t) , without use of Lenz's Law" : (6) BTpdv(POIo(t),t) = BTodv(POIo,t)] = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART)/Rpcs(POIo(t),t)^2 - EIods(POIo(t),t) ] where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) in RFp coordinates /*OOOPS!!! - I should have taken : /% (9) BTpdv(POIo(t),t) = BTodv(POIo,t) = Vons(PART)/c *Rocs(POIo)*sin(Aθoc(POIo)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) - EIods(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ] where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Anyways, still proceeding from (6)* : ∂[∂(t): BTodv(POIo,t)] = ∂[∂(t): Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t))*[Q(PART)/Rpcs(POIo(t),t)^2 - EIods(POIo(t),t)]] = ∂[∂(t): Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t))] *[ Q(PART)/Rpcs(POIo(t),t)^2 - EIods(POIo(t),t) ] + Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *∂[∂(t): Q(PART)/Rpcs(POIo(t),t)^2 - EIods(POIo(t),t)] = ∂[∂(t): Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t))] *[ Q(PART)/Rpcs(POIo(t),t)^2 - EIods(POIo(t),t) ] + Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ ∂[∂(t): Q(PART)/Rpcs(POIo(t),t)^2] - ∂[∂(t): EIods(POIo(t),t)] ] (1) = Vons(PART)/c* { ∂[∂(t): sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t))] *[ Q(PART) /Rpcs(POIo(t),t)^2 - EIods(POIo(t),t) ] + sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART)*∂[∂(t): 1/Rpcs(POIo(t),t)^2]- ∂[∂(t): EIods(POIo(t),t)] ] } Removing Rodh(Vonv_X_Rpcv(POIo(t),t)) from within derivatives as a constant, Equation (1) becomes : (1) BTodv(POIo,t)] = Vons(PART)/c* { ∂[∂(t): sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t))] *[ Q(PART) /Rpcs(POIo(t),t)^2 - EIods(POIo(t),t) ] + sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART)*∂[∂(t): 1/Rpcs(POIo(t),t)^2] - ∂[∂(t): EIods(POIo(t),t)] ] } (3) = Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t))* { ∂[∂(t): sin(Aθpc(POIo(t),t))] *[ Q(PART) /Rpcs(POIo(t),t)^2 - EIods(POIo(t),t) ] + sin(Aθpc(POIo(t),t)) *[ Q(PART)*∂[∂(t): 1/Rpcs(POIo(t),t)^2] - ∂[∂(t): EIods(POIo(t),t)] ] } /*+--+ From "∂[∂(t): sin(Aθpc(POIo(t),t)) ]" : (1) ∂[∂(t): sin(Aθpc(POIo(t),t))] = -Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) * [ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] / [ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART) *t)^2 ]^(3/2) Looking at : (4) ∂[∂(t): 1/Rpcs(POIo(t),t)^2 ] = ∂[∂(t): Rpcs(POIo(t),t)^(-2) ] = -2*Rpcs(POIo(t),t)^(-3)*∂[∂(t): Rpcs(POIo(t),t)] From "∂[∂(t): Rpcs(POIo(t),t) ] = ∂[∂(t): |Rpcv(POIo(t),t)|]" : (7) ∂[∂(t): Rpcs(POIo(t),t)] = Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(1/2) /*From "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)" : /% (2) Rpcs(POIo(t),t)) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (7)* & (2)* into (4) : (4) ∂[∂(t): 1/Rpcs(POIo(t),t)^2 ] = ∂[∂(t): Rpcs(POIo(t),t)^(-2) ] = -2*Rpcs(POIo(t),t)^(-3)*∂[∂(t): Rpcs(POIo(t),t)] = -2*{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^(-3) *Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(1/2) = -2*Vons(PART) *{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-3/2) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(1/2) (5) = -2*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 /*+--+ Subbing (1)*,(5)*,(5) into (3) (3) ∂[∂(t): BTodv(POIo,t)] = Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo))* {∂[∂(t): sin(Aθpc(POIo(t),t)) ] *[ Q(PART)/Rpcs(POIo(t),t)^2 - EIods(POIo(t),t) ] + sin(Aθpc(POIo(t),t)) *[ Q(PART)*∂[∂(t): 1/Rpcs(POIo(t),t)^2 ] - ∂[∂(t): EIods(POIo(t),t)] ] } = Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo))* { - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(3/2) *[ Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^2 - EIods(POIo(t),t) ] + sin(Aθpc(POIo(t),t)) *[ Q(PART)*-2*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 - ∂[∂(t): EIods(POIo(t),t)] ] } Summarizing : (mathH)/* Note that I keep the direction vector Rodh(Vonv_X_Rpcv(POIo)) for simplicity!! /% ∂[∂(t): BTodv(POIo,t)] = Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo))* { - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(3/2) *[ Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^2 - EIods(POIo(t),t) ] + sin(Aθpc(POIo(t),t)) *[ Q(PART)*-2*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 - ∂[∂(t): EIods(POIo(t),t)] ] } (endMath) /*+--+ Limit checks : +-+ Dimensional consistency, as a convention here take [unit vectors, c] as dimensionless : (see file "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt" sub-sub-section "Dimensional analysis (Gaussian units)" for dimensional analysis notes) /% (6) ∂[∂(t): BTodv(POIo,t)] = Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t))* { - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(3/2) *[ Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } + EIods(POIo(t),t) ] + sin(Aθpc(POIo(t),t)) *[ Q(PART)*-2*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 + ∂[∂(t): EIods(POIo(t),t)] ] } /* = (length/time) { - (length)*(length/time) *[ - (length) + (length/time)*(time) ] /[ (length)^2 - (length)*(length/time)*(time) + (length/time*time)^2 ]^(3/2) *[ (charge) /{ (length)^2 - (length)*(length/time)*(time) + [(length/time)*(time)]^2 } + (charge/length^2) ] + 1 *[ (charge)*-(length/time) *[ - (length) + (length/time)*(time) ] /{ (length)^2 - (length)*(length/time)*(time) + [(length/time)*(time)]^2 }^2 + (charge/length^2)/(time) ] } = (length/time) { - (length)^2/(time) *[ - (length) + (length) ] /[ (length)^2 - (length)^2 + (length)^2 ]^(3/2) *[ (charge) /{ (length)^2 - (length)^2 + (length)^2 } + (charge/length^2) ] + 1 *[ (charge)*-(length/time) *[ - (length) + (length) ] /{ (length)^2 - (length)^2 + (length)^2 }^2 + (charge/length^2/time) ] } = (length/time) { (length)^2/(time)*(length)/[(length)^2]^(3/2) *[ (charge)/(length)^2 + (charge/length^2) ] + [ (charge)*(length/time)*(length) /(length)^2^2 + (charge/length^2/time) ] } = (length/time) { (length)^(2+1-3)/(time) * (charge)/(length)^2 + [ (charge)*(length)^(1+1-4)/(time) + (charge/length^2/time) ] } = (length/time) { charge/length^2/time - [ charge/length^2/time + charge/length^2/time ] } = charge/length/time^2 From "Dimensional analysis (Gaussian units) ", sub-section "Dimensional analysis (Gaussian units)" : units of ∂[∂(t): BTpdv(POIo(t),t)] = (charge/length/time^2) units OK as all units reduce to (charge/length/time^2) (ignoring permittivity & permeability for Gaussian units) +-+ Reconciliation of RFo and RFp results later .... /********************************************** >>>>>> Lenz's Induction Law - Basics and Calculus 23Mar2016 - Equations were all renumbered, augmented by new equations. Need to change references to these equations in the rest of this document. <<< 24Mar2018 – My vector expressions are WRONG in that vector components must be reversed, not just negated!!! >>> However, the same expression results for simple vector additions, but does this work for [inner, cross] product and other functions? I doubt it... or at least I would have to prove it. IE - In vector notation, does the negative sign imply opposite directions? /********************* >>>>>>>>> Lenz's Induction Law and it's context From Lucas's book, p64h0.4 Equation (4-5) : /%(4-5) EI_LENZodv(POIo,t) ∝ E0ods(POIo,t) * (- Rpch(POIo(t),t) ) → 24Mar2018 changed from E0odv(POIo,t) where ET_LENZodv(POIo) = E0odv(POIo) + EI_LENZodv(POIo) /*From Lucas's book, p70h0.85 Equation (4-31) : → <<< 24Mar2018 YESSS!!! Lucas had that right >>> /% (mathH)/* (4-31) /% EI_LENZods(POIo,t)|(Aθpc(POIo(t),t=0))*Rpch(POIo(t),t) = -λ(Vons(PART))*E0ods(POIo)*Rpch(POIo(t),t) (endMath) where λ(Vons(PART)) is a positive real constant, given that Vonv(PART) is a constant /*<<< It's not clear to me WHY this has been restricted to the condition |(Aθpc(POIo))=0), but in the developments below I will ignore that restiction. >>> /*+-----+ 15May2016 From : 30Mar2016 ?Inconsistency? : Lenz's Induction Law versus Barnes iterations - I have been assuming : ET = E0 + EI = E0 - lambda(v)*E0 = (1 - lambda(v))*E0 = (1 - beta^2)*E0 but in the end Lucas uses : ET = E0*(1-beta^2)/(1 - beta^2*sin(theta)^2)^(3/2) This is not consistent!!! see (4-31) to (4-43) The denominator of that last expression = 1 when (Aθpc(POIo))=0), so perhaps this explains the condition used by Lucas. /*+-----+ /%(1) EI_LENZodv(POIo(t),t) = -λ(Vons(PART))*E0odv(POIo,t) where λ(Vons(PART)) is a positive real function of speed /*Note that in Chapter 4 of Lucas's book, [Q(PART), Vonv(PART), lambda(v), ???] are all constant. For a Point Of Interest (POIo) in the observer reference frame (RFo), ???? AT CONSTANT RELATIVE VELOCITY (POIo, particle) by Lenz's Law (see (4-31)*, then [EIodv(POIo,t),∂[∂(t): EIodv(POIo,t)]] are just scalar multiples of [E0odv(POIo,t),∂[∂(t): E0odv(POIo,t)]]. 12May2016 Remember, this ONLY applies to the constant particle velocity situation as in Chapter 4!!! 12May2016 NOT generally correct!! -> sin(theta) must be taken into account at the surface of a particle, even though here I don't need to as only point-sized particles are being addressed at this stage, so vector EIodv(POIo,t) is just a scalar multiple of EIodv(POIo,t). However, this issue does become a problem for finite-sized particles. 12May2016 p68h0.5 Figure (4-4) Halelujeh!! -- Sometimes, Lucas's angle theta is the angle at the surface of a charged particle between the suface normal vector and B !!! /********************* /%>>>>>>>>> EI_LENZodv(POIo,t) = EI_LENZpdv(POIo(t),t), using Lenz's Induction Law /*From "Lenz's Induction Law and it's context" : /% (1)* EI_LENZodv(POIo(t),t) = -λ(Vons(PART))*E0odv(POIo,t) where λ(Vons(PART)) is a positive real function of speed /*+-----+ (RFp) basis /%From "E0odv(POIo,t) = E0pdv(POIo(t),t)" : (1)** E0pdv(POIo(t),t) = E0odv(POIo,t) = Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) /*Subbing (1)** into (1)* : /% (mathH) EI_LENZpdv(POIo(t),t) = -λ(Vons(PART))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) (endMath) /* where lambda(v) is a positive real function of speed /*+-----+ (RFo) basis From "Rpcs(POIo(t),t)" : /% (2) Rpcs(POIo(t),t) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (2)* into (2) : EI_LENZodv(POIo(t),t) = EI_LENZpdv(POIo(t),t) = -λ(Vons(PART))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) = -λ(Vons(PART))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^2 Finally : (mathH) EI_LENZodv(POIo(t),t) = -λ(Vons(PART))*Q(PART)*Rpch(POIo(t),t) /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2} (endMath) /********************* >>>>>>>>> ET_LENZodv(POIo(t),t) = ET_LENZpdv(POIo(t),t), using Lenz's Induction Law /*+-----+ (RFp) basis Following Lucas p67h0.6 Eqn (4-13) & (4-41) : /% (1)* ET_LENZpdv(POIo(t),t) = E0pdv(POIo(t),t) + EI_LENZpdv(POIo(t),t) From "E0odv(POIo,t) = E0pdv(POIo(t),t)" : (1)** E0pdv(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) From "EI_LENZodv(POIo(t),t) = EI_LENZpdv(POIo(t),t), using Lenz's Induction Law" : (2)* EI_LENZpdv(POIo(t),t) = -λ(Vons(PART))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) Subbing (1)** & (2)* into (1)* : (1)* ET_LENZpdv(POIo(t),t) = E0pdv(POIo(t),t) + EI_LENZpdv(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) + -λ(Vons(PART))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) = (1 - λ(Vons(PART)))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) Summarizing : (mathH) ET_LENZpdv(POIo(t),t) = (1 - λ(Vons(PART)))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) (endMath) /*+-----+ (RFo) basis Following Lucas p67h0.6 Eqn (4-13) & (4-41) : /% (1)* ET_LENZodv(POIo(t),t) = E0odv(POIo,t) + EI_LENZodv(POIo(t),t) From "E0odv(POIo,t) = E0pdv(POIo(t),t)" : (5)* E0odv(POIo) = Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } where Rpch(POIo(t),t) = Rodh(POIo) = displacement vector [start : (POIo), length : 1, theta : arccos(Aθpc(POIo(t),t)), phi : Aφoc(POIo)] and cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) From "EI_LENZodv(POIo(t),t) = EI_LENZpdv(POIo(t),t), using Lenz's Induction Law" : (3)* EI_LENZodv(POIo(t),t) = -λ(Vons(PART))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } Subbing (5)* & (3)* into (1)* : (1)* ET_LENZodv(POIo(t),t) = E0odv(POIo,t) + EI_LENZodv(POIo(t),t) = Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } + -λ(Vons(PART))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } = (1 - λ(Vons(PART)))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } So : (mathH) ET_LENZodv(POIo(t),t) = (1 - λ(Vons(PART)))*Q(PART)*Rpch(POIo(t),t) /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2} (endMath) /********************* >>>>>>>>> EI_LENZods(POIo(t),t) = EI_LENZpds(POIo(t),t), using Lenz's Induction Law (1) EI_LENZods(POIo(t),t) = EI_LENZpds(POIo(t),t) = |EI_LENZodv(POIo(t),t)| = |EI_LENZpdv(POIo(t),t)| For now - ARBITRARY sign on equations for scalar derivatives, as I have to go back & fix the deerivations (not as straightforward as it sounds!). ∂[∂(t): EI_LENZods(POIo(t),t)] has the opposite sign of ∂[∂(t): E0ods(POIo,t)] Note that for : /% -PI/2 <= Aθpc(POIo(t),t)) <= PI/2 : then cos(Aθpc(POIo(t),t)) >0 and ∂[∂(t): E0pds(POIo(t),t)] < 0, and ∂[∂(t): EI_LENZpds(POIo(t),t)] < 0 /*19Dec2017 The following is nonsensical, as PI/2 >= -PI/2 !!?? /% PI/2 <= Aθpc(POIo(t),t)) <= -PI/2 : then cos(Aθpc(POIo(t),t)) <0 and ∂[∂(t): E0pds(POIo(t),t)] > 0, and ∂[∂(t): EI_LENZpds(POIo(t),t)] > 0 /*For Vonv(PART) > 0 (as per definition of BOTH coordinate frames of reference!), these results make sense /*+-----+ (RFp) basis /%From "EI_LENZodv(POIo(t),t) = EI_LENZpdv(POIo(t),t), using Lenz's Induction Law" : 2*) EI_LENZpdv(POIo(t),t) = -λ(Vons(PART))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) /*Subbing (2)* into (1), /% EI_LENZods(POIo(t),t) = |-λ(Vons(PART))*Q/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t)| /* Absolute values of multiplicataion/division series - simply do the absolute value of each term /* Absolute values of addition/subtraction series - cannot just do individual terms - must take whole series I need to fix this!! : From "Scalar absolute values, [vector, matrix] norms - simplification of expressions" : scalar norms - if all terms are scalars - multiplicative/divisive : |product(x1,x2,x3,...)| = product(|x1|,|x2|,|x3|,...) - subtraction/ addition : |sum (x1,x2,x3,...)| ≠ ≠ sum/diff(|x1|,|x2|,|x3|,...) in general vector norms - if some terms are vectors - multiplicative/divisive : |product(x1,x2,x3,...)| = product(|x1|,|x2|,|x3|,...) /% EI_LENZods(POIo(t),t) = |-λ(Vons(PART))*Q/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t)| = |-λ(Vons(PART))|*|Q|/|Rpcs(POIo(t),t)^2|*|Rpch(POIo(t),t)| /*As we know that : - Rpch(POIo(t),t) is a unit vector, so we can replace it by |Rpch(POIo(t),t)| = 1 - lambda(v) is positive (?) - Q(PART) can be positive or negative real, so just tack the norm /% (mathH) EI_LENZpds(POIo(t),t) = λ(Vons(PART))*|Q(PART)|/Rpcs(POIo(t),t)^2 (endMath) /*+-----+ (RFo) basis /%From "Rpcs(POIo(t),t)" : (2) Rpcs(POIo(t),t) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (2)* into (2) : EI_LENZods(POIo(t),t) = EI_LENZpds(POIo(t),t) = λ(Vons(PART))*Q(PART)/Rpcs(POIo(t),t)^2 = λ(Vons(PART))*Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^2 Summarizing : (mathH) EI_LENZods(POIo(t),t) = EI_LENZpds(POIo(t),t) = λ(Vons(PART))*|Q(PART)| /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2} (endMath) /********************* >>>>>>>>> ET_LENZods(POIo(t),t) = ET_LENZpds(POIo(t),t), using Lenz's Induction Law IMPORTANT NOTE : There is an explicit belief in mainstream physics, AND IN LUCAS'S book, an implicit assumption, that speeds can never exceed the speed of light in a vacuum, c. However, that should NOT be a limitation of Lucas's Universal force!! There is therefore no guarantee that v/c is <1, and therefore that : /% Nyet : 0 <= [(1 - λ(Vons(PART))), (1 - β^2) <= 1 /*This affects derivatives! /*+-----+ (RFp) basis /%From "ET_LENZodv(POIo(t),t) = ET_LENZpdv(POIo(t),t), using Lenz's Induction Law" : (1)* ET_LENZpdv(POIo(t),t) = (1 - λ(Vons(PART)))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) By definition : ET_LENZpds(POIo(t),t) = |ET_LENZpdv(POIo(t),t)| = |(1 - λ(Vons(PART)))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t)| = |(1 - λ(Vons(PART)))|*|Q(PART)|/|Rpcs(POIo(t),t)^2|*|Rpch(POIo(t),t)| But [λ(Vons(PART)),Rpcs(POIo(t),t),Rpch(POIo(t),t)] are positive real (& in any case Rpcs(POIo(t),t) is squared). And |Rpch(POIo(t),t)| = 1. So : (mathH) ET_LENZpds(POIo(t),t) = |(1 - λ(Vons(PART)))|*|Q(PART)|/Rpcs(POIo(t),t)^2 (endMath) /*+-----+ (RFo) basis /%From "ET_LENZodv(POIo(t),t) = ET_LENZpdv(POIo(t),t), using Lenz's Induction Law" : (2)* ET_LENZodv(POIo(t),t) = (1 - λ(Vons(PART)))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } /*By definition : /% ET_LENZods(POIo(t),t) = |ET_LENZodv(POIo(t),t)| = | (1 - λ(Vons(PART)))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } | = |(1 - λ(Vons(PART)))|*|Q(PART)|*|Rpch(POIo(t),t)| /|{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }| But [λ(Vons(PART)),Rpcs(POIo(t),t),Rpch(POIo(t),t)] are positive real (& in any case Rpcs(POIo(t),t) is squared). And |Rpch(POIo(t),t)| = 1. So : (mathH) ET_LENZods(POIo(t),t) = |ET_LENZodv(POIo(t),t)| = |(1 - λ(Vons(PART)))|*|Q(PART)| /|{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}| (endMath) /********************* >>>>>>>>> BT_LENZodv(POIo(t),t) = BT_LENZpdv(POIo(t),t), using Lenz's Induction Law Here I will simply adapt results in "BT_LENZodv(POIo(t),t) = BT_LENZpdv(POIo(t),t), without use of Lenz's Law". /*+-----+ (RFp) basis /%From "BT_LENZodv(POIo(t),t) = BT_LENZpdv(POIo(t),t), without use of Lenz's Law" : (6) BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t) = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART)/Rpcs(POIo(t),t)^2 + EI_LENZpds(POIo(t),t) ] where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) From "EI_LENZodv(POIo(t),t) = EI_LENZpdv(POIo(t),t), using Lenz's Induction Law" : (2) EIodv(POIo) = EI_LENZpdv(POIo(t),t) = -λ(Vons(PART))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) Subbing (2)* into (6)* : BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t) = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART)/Rpcs(POIo(t),t)^2 + EI_LENZpds(POIo(t),t) ] = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART)/Rpcs(POIo(t),t)^2 - λ(Vons(PART))*Q(PART)/Rpcs(POIo(t),t)^2 ] = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *Q(PART)/Rpcs(POIo(t),t)^2*(1 - λ(Vons(PART))) Summarizing : (mathH) BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t) = (1-λ(Vons(PART)))*Q(PART)*Vons(PART)/c*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^2*Rodh(Vonv_X_Rpcv(POIo(t),t)) (endMath) where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) /*+-----+ (RFo) basis /%From "Rpcs(POIo(t),t)" : (2) Rpcs(POIo(t),t) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) From "sin(Aθpc(POIo(t),t))" : (5) sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (2)*,(5)* into (1) : (1) BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t) = [1-λ(Vons(PART))]*Q(PART)*Vons(PART)/c*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^2*Rodh(Vonv_X_Rpcv(POIo(t),t)) = [1-λ(Vons(PART))]*Q(PART)*Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) *sin(Aθpc(POIo(t),t)) /Rpcs(POIo(t),t)^2 = [1-λ(Vons(PART))]*Q(PART)*Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) *Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^2 Summarizing : (mathH)/* is there a mistake here? /% BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t) = [1-λ(Vons(PART))]*Q(PART)*Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) *Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) (endMath) /********************* >>>>>>>>> ∂[∂(t): EI_LENZodv(POIo(t),t)] = ∂[∂(t): EI_LENZpdv(POIo(t),t)] - using Lenz's Law, proper vector approach /%From "EI_LENZodv(POIo(t),t) = EI_LENZpdv(POIo(t),t), using Lenz's Induction Law" : (1) EIodv(POIo) = EI_LENZpdv(POIo(t),t) = -λ(Vons(PART))*E0odv(POIo) Therefore : ∂[∂(t): EIodv(POIo)] = ∂[∂(t): EI_LENZpdv(POIo(t),t)] = ∂[∂(t): - λ(Vons(PART))*E0odv(POIo)] Or : (mathH) ∂[∂(t): EIodv(POIo)] = ∂[∂(t): EI_LENZpdv(POIo(t),t)] = -λ(Vons(PART))*∂[∂(t): E0odv(POIo)] (endMath) /*+-----+ (RFp) basis /%From "∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)]" : (6) ∂[∂(t): E0pdv(POIo(t),t)] = Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[ sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) ] where : Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 Multiplying (6)* by -λ(Vons(PART)) : ∂[∂(t): EI_LENZpdv(POIo(t),t) = -λ(Vons(PART))*∂[∂(t): E0pdv(POIo(t),t)] Yielding : (mathH)/* where : λ(Vons(PART)) is a positive constant, Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t), RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 /% ∂[∂(t): EI_LENZpdv(POIo(t),t)] = -λ(Vons(PART))*Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t)] (endMath) /*+-----+ (RFo) basis /%From "∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)]" : (9)* ∂[∂(t): E0pdv(POIo(t),t)] = Q(PART)*Vons(PART) *[ Rocs(POIo)*sin(Aθoc(POIo)) *RDEpdh(POIo(t),∂(t)) + 2*[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rpch(POIo(t),t) ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 where : Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 Aθpc(POIo(t),t) = arcsin [ Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ] = arccos [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) ] Therefore, multiplying (9)* by -λ(Vons(PART)) : (1) ∂[∂(t): EI_LENZodv(POIo(t),t) = -λ(Vons(PART))*∂[∂(t): E0pdv(POIo(t),t)] from "sin(Aθpc(POIo(t),t))" : (5)* sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) from "cos(Aθpc(POIo(t),t))" : (1)* cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) Subbing (1),(5)*,(1)* into (9)* (9)* ∂[∂(t): EI_LENZodv(POIo(t),t)] = -λ(Vons(PART))*Q(PART)*Vons(PART) *[ Rocs(POIo)*sin(Aθoc(POIo)) *RDEpdh(POIo(t),∂(t)) + 2*[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rpch(POIo(t),t) ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 So : (mathH)/* where : λ(Vons(PART)) is a positive constant, Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t), RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 /% ∂[∂(t): EI_LENZodv(POIo(t),t)] = -λ(Vons(PART))*Q(PART)*Vons(PART) *[ Rocs(POIo)*sin(Aθoc(POIo)) *RDEpdh(POIo(t),∂(t)) + 2*[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rpch(POIo(t),t) ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 (endMath) /********************* /%>>>>>>>>> ∂[∂(t): EI_LENZods(POIo(t),t)] = ∂[∂(t): EI_LENZpds(POIo(t),t)] - using Lenz's Law, based on proper vector approach /*+-----+ (RFp) basis /%From "∂[∂(t): EI_LENZodv(POIo(t),t)] = ∂[∂(t): EI_LENZpdv(POIo(t),t)] - using Lenz's Law, proper vector approach" : (2)* ∂[∂(t): EI_LENZpdv(POIo(t),t)] = -λ(Vons(PART))*∂[∂(t): E0pdv(POIo(t),t)] = -λ(Vons(PART))*Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[ sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) ] where : λ(Vons(PART)) is a positive constant Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 /*But this is not used directly in this "proper vector approach", for which a rederivation is necessary to be sure!!! /%From "EI_LENZodv(POIo(t),t) = EI_LENZpdv(POIo(t),t), using Lenz's Induction Law" : (2)** EIodv(POIo) = EI_LENZpdv(POIo(t),t) = -λ(Vons(PART))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) Subbing (2)** into definition of ∂[∂(t): EI_LENZods(POIo(t),t)] : (1) ∂[∂(t): EI_LENZods(POIo(t),t)] = ∂[∂(t): |EI_LENZodv(POIo(t),t)|] = ∂[∂(t): | - λ(Vons(PART))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t)|] (1) = λ(Vons(PART))*|Q(PART)|*∂[∂(t): |Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2|] From "∂[∂(t): E0ods(POIo,t)] = ∂[∂(t): E0pds(POIo(t),t)] - proper E0odv(POIo,t) vector approach" : /* Key intermediate result useful elsewhere : /% (12)* ∂[∂(t): |Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2|] = 2*cos(Aθpc(POIo(t),t))*Vons(PART)/Rpcs(POIo(t),t)^3 /*Subbing (12)* into (1) : /% (1) ∂[∂(t): EI_LENZods(POIo(t),t)] = ∂[∂(t): |EI_LENZodv(POIo(t),t)|] = λ(Vons(PART))*|Q(PART)|*∂[∂(t): |Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2|] = λ(Vons(PART))*|Q(PART)|*2*cos(Aθpc(POIo(t),t))*Vons(PART)/Rpcs(POIo(t),t)^3 /*For now - ARBITRARY sign on equations for scalar derivatives, as I have to go back & fix the derivations (not as straightforward as it sounds!). /% ∂[∂(t): EI_LENZods(POIo(t),t)] has the opposite sign of ∂[∂(t): E0ods(POIo,t)] /*Note that for : /% -PI/2 <= Aθpc(POIo(t),t)) <= PI/2 : then cos(Aθpc(POIo(t),t)) >0 and ∂[∂(t): E0pds(POIo(t),t)] > 0, and ∂[∂(t): EI_LENZpds(POIo(t),t)] > 0 /*19Dec2017 The following is nonsensical, as PI/2 >= -PI/2 !!?? /% PI/2 <= Aθpc(POIo(t),t)) <= -PI/2 : then cos(Aθpc(POIo(t),t)) <0 and ∂[∂(t): E0pds(POIo(t),t)] > 0, and ∂[∂(t): EI_LENZpds(POIo(t),t)] > 0 /*For Vonv(PART) > 0 (as per definition of BOTH coordinate frames of reference!), these results make sense Summarizing : /% (mathH) ∂[∂(t): EI_LENZpds(POIo(t),t)] = 2*λ(Vons(PART))*|Q(PART)|*cos(Aθpc(POIo(t),t))*Vons(PART)/Rpcs(POIo(t),t)^3 (endMath) /*+-----+ (RFo) basis /%From "Rpcs(POIo(t),t)" : (3)* Rpcs(POIo(t),t) = |Rpcv(POIo(t),t)| = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) From "cos(Aθpc(POIo(t),t))" : (1)* cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) Subbing (3)* & (1)* into (2) : (2) ∂[∂(t): EI_LENZods(POIo(t),t)] = 2*λ(Vons(PART))*|Q(PART)|*cos(Aθpc(POIo(t),t))*Vons(PART)/Rpcs(POIo(t),t)^3 = 2*λ(Vons(PART))*|Q(PART)|*Vons(PART) *cos(Aθpc(POIo(t),t)) /Rpcs(POIo(t),t)^3 = 2*λ(Vons(PART))*|Q(PART)|*Vons(PART) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^3 = 2*λ(Vons(PART))*|Q(PART)|*Vons(PART) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^2 Summarizing : (mathH) ∂[∂(t): EI_LENZods(POIo(t),t)] = 2*λ(Vons(PART))*|Q(PART)|*Vons(PART) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^2 (endMath) /********************* /%>>>>>>>>> ∂[∂(t): EI_LENZods(POIo(t),t)] = ∂[∂(t): EI_LENZpds(POIo(t),t)] - using Lenz's Law, cheating non-vector derivative of E0ods(POIo,t) /*+-----+ (RFp) basis From "Lenz's Induction Law and it's context" : /% (6)* ∂[∂(t): EI_LENZods(POIo(t),t)] = λ(Vons(PART))*∂[∂(t): E0ods(POIo)] /*+--+ From "E0ods(POIo,t) = E0pds(POIo(t),t)" : /% (1)* E0ods(POIo,t) = Q(PART)/Rpcs(POIp)^2 /*Taking the "cheating direct non-vector derivative" of (1)* : /% ∂[∂(t): E0ods(POIo,t)] = ∂[∂(t): Q(PART)/Rpcs(POIp)^2] = Q(PART)*∂[∂(t): 1/Rpcs(POIp)^2] (1) = Q(PART)*-2*Rpcs(POIp)^(-3)*∂[∂(t): Rpcs(POIp)] From "∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|]" : (1)** ∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|] = -Vons(PART)*cos(Aθpc(POIo(t),t)) Subbing (1)** into (1) : ∂[∂(t): E0ods(POIo,t)] = Q(PART)*-2*Rpcs(POIp)^(-3)*∂[∂(t): Rpcs(POIp)] = Q(PART)*-2*Rpcs(POIp)^(-3)*-Vons(PART)*cos(Aθpc(POIo(t),t)) (2) = 2*Q(PART)*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIp)^3 Subbing (2) into (6)* : (mathH) ∂[∂(t): E0ods(POIo,t)] = 2*λ(Vons(PART))*Q(PART)*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIp)^3 (endMath) /*+-----+ LIMIT CHECKS : From "Dimensional analysis (Gaussian units)" in file "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt" : E units = (charge/length^2) therefore ∂[∂(t): E] -> (charge/length^2/time) Dimensional consistency - OK, as RHS & LHS reduce to (charge/length^2/time). /%Comparing current result (2) to "∂[∂(t): EI_LENZods(POIo(t),t)] = ∂[∂(t): EI_LENZpds(POIo(t),t)] - using Lenz's Law, based on proper vector approach" (2)* : (2) ∂[∂(t): E0ods(POIo,t)] = -2*λ(Vons(PART))* Q(PART) *Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIp)^3 (2)* ∂[∂(t): EI_LENZods(POIo(t),t)] = 2*λ(Vons(PART))*|Q(PART)|*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 /*OTHER THAN the minus sign and |Q| (which should be obvious anyways), this is OK, so a "proper vector approach" was NOT actually necessary, but it is helpful for error trapping. /*+-----+ (RFo) basis Because (3) above is the same as for the "proper vector approach", there is no need to repeat exactly the same derivation here. /********************* >>>>>>>>> ∂[∂(t): ET_LENZodv(POIo(t),t)] = ∂[∂(t): ET_LENZpdv(POIo(t),t)], using Lenz's Induction Law /*+-----+ (RFp) basis /%From "ET_LENZodv(POIo(t),t) = ET_LENZpdv(POIo(t),t), using Lenz's Induction Law" : (1)* ET_LENZpdv(POIo(t),t) = (1 - λ(Vons(PART)))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) So : ∂[∂(t): ET_LENZpdv(POIo(t),t)] = ∂[∂(t): (1 - λ(Vons(PART)))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t)] For Chapter 4, [(1 - λ(Vons(PART))),Q(PART)] are constants with respect to (wrt) time. So : .1) ∂[∂(t): ET_LENZpdv(POIo(t),t)] = (1 - λ(Vons(PART)))*Q(PART) *∂[∂(t): Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2] = (1 - λ(Vons(PART)))*Q(PART) *{ ∂[∂(t): Rpch(POIo(t),t)] / Rpcs(POIo(t),t)^2 + Rpch(POIo(t),t) *∂[∂(t): Rpcs(POIo(t),t)^( - 2)] } From "∂[∂(t): Rpch(POIo(t),t)]" : (2)* ∂[∂(t): Rpch(POIo(t),t)] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*RDEpdh(POIo(t),t) 22Jan2017 that's NOT the expression? where : λ(Vons(PART)) is a positive constant Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),t) is anchored at end of Rpch(POIo(t),t) and is at angle Aθpc(POIo(t),t) + PI/2, ie perpendicular to Rpch(POIo(t),t) angle Aφpc(POIo(t),t) doesn't change Looking at ∂[∂(t): Rpcs(POIo(t),t)^( - 2)] : (2) ∂[∂(t): Rpcs(POIo(t),t)^( - 2)] = (-2)*Rpcs(POIo(t),t)^(-3)*∂[∂(t): Rpcs(POIo(t),t)] From "∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|]" : (1)** ∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|] = -Vons(PART)*cos(Aθpc(POIo(t),t)) Subbing (1)** into (2) : (3) ∂[∂(t): Rpcs(POIo(t),t)^( - 2)] = (-2)*Rpcs(POIo(t),t)^(-3)*∂[∂(t): Rpcs(POIo(t),t)] = (-2)*Rpcs(POIo(t),t)^(-3)*-Vons(PART)*cos(Aθpc(POIo(t),t)) = 2*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 Subbing (2)* & (3) into (1) : (1) ∂[∂(t): ET_LENZpdv(POIo(t),t)] = (1 - λ(Vons(PART)))*Q(PART) *{ ∂[∂(t): Rpch(POIo(t),t)] / Rpcs(POIo(t),t)^2 + Rpch(POIo(t),t) *∂[∂(t): Rpcs(POIo(t),t)^( - 2)] } = (1 - λ(Vons(PART)))*Q(PART) *{ Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) *RDEpdh(POIo(t),t) / Rpcs(POIo(t),t)^2 + 2*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 *Rpch(POIo(t),t) } = (1 - λ(Vons(PART)))*Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *{ sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),t) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) } Summarizing : (mathH)/* Note that RDEpdh(POIo(t),t) & Rpch(POIo(t),t) are NOT the same unit vector! /% ∂[∂(t): ET_LENZpdv(POIo(t),t)] = (1 - λ(Vons(PART)))*Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *{sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),t) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t)} (endMath) /*+-----+ LIMIT CHECKS : From "Dimensional analysis (Gaussian units)" in file "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt" : ∂[∂(t): E] units = charge/length^2/time /%(4) ∂[∂(t): ET_LENZpdv(POIo(t),t)] (charge/length^2/time) = (1 - λ(Vons(PART)))*Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 charge length/time length^(-3) *{ sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),t) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) } /* (all dimensionless) = (charge/length^2/time) Dimensional consistency - OK, as RHS & LHS reduce to (charge/length^2/time). /%Equation (4) is (1 - λ(Vons(PART))) times "∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)]" Equation (6), as expected. /*+-----+ (RFo) basis (re-starting equation numbers) /%From "ET_LENZodv(POIo(t),t) = ET_LENZpdv(POIo(t),t), using Lenz's Induction Law" : (2)* ET_LENZodv(POIo(t),t) = (1 - λ(Vons(PART)))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } So : ∂[∂(t): ET_LENZodv(POIo(t),t)] = ∂[∂(t): (1 - λ(Vons(PART)))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } ] For Chapter 4, [(1 - λ(Vons(PART))),Q(PART)] are constants with respect to (wrt) time. = (1 - λ(Vons(PART)))*Q(PART) *∂[∂(t): Rpch(POIo(t),t)/{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}] (1) = (1 - λ(Vons(PART)))*Q(PART) *{ ∂[∂(t): Rpch(POIo(t),t)] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } + Rpch(POIo(t),t)*∂[∂(t): 1/{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}] } From "∂[∂(t): Rpch(POIo(t),t)]" : (2)* ∂[∂(t): Rpch(POIp)] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*RDEpdh(POIo(t),t) where RDEpdh(POIo(t),t) is anchored at end of Rpch(POIo(t),t) and is at angle Aθpc(POIo(t),t) + PI/2, ie perpendicular to Rpch(POIo(t),t) angle Aφpc(POIo(t),t) doesn't change Looking at : ∂[∂(t): 1/{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}] = (-1)*{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-2) *∂[∂(t): Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2] For Chapter 4, [Rocs(POIo),cos(Aθoc(POIo)),Vons(PART)] are constants with respect to (wrt) time. = (-1)*{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-2) *{ - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART) + Vons(PART)^2*2*t} = (-1)*{ - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART) + Vons(PART)^2*2*t } *{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-2) (2) = (-2)*{ - Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART) + Vons(PART)^2*t } /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 Subbing (2)* & (2) into (1) : (1) ∂[∂(t): ET_LENZodv(POIo(t),t)] = (1 - λ(Vons(PART)))*Q(PART) *{ ∂[∂(t): Rpch(POIo(t),t)] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } + Rpch(POIo(t),t)*∂[∂(t): 1/{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}] } = (1 - λ(Vons(PART)))*Q(PART) *{ Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*RDEpdh(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } + Rpch(POIo(t),t) * (-2)*{ - Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART) + Vons(PART)^2*t } /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 } (3) = (1 - λ(Vons(PART)))*Q(PART) *{ Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*RDEpdh(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } - 2*Vons(PART)*Rpch(POIo(t),t) *{ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t } /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART) *t]^2 }^2 } From "Rpcs(POIo(t),t)" : (3)* Rpcs(POIo(t),t) = |Rpcv(POIo(t),t)| = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) from "sin(Aθpc(POIo(t),t))" : (5)* sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (3)* & (5)* into (3) : (3) ∂[∂(t): ET_LENZodv(POIo(t),t)] = (1 - λ(Vons(PART)))*Q(PART) *{ Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*RDEpdh(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } - 2*Vons(PART)*Rpch(POIo(t),t) *{ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t } /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART) *t]^2 }^2 } = (1 - λ(Vons(PART)))*Q(PART) *{ Vons(PART)*RDEpdh(POIo(t),t) *sin(Aθpc(POIo(t),t)) /Rpcs(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } - 2*Vons(PART)*Rpch(POIo(t),t) *{ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t } /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART) *t]^2 }^2 } = (1 - λ(Vons(PART)))*Q(PART)*Vons(PART) *{ RDEpdh(POIo(t),t) * Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } - 2*Rpch(POIo(t),t) *{ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t } /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART) *t]^2 }^2 } = (1 - λ(Vons(PART)))*Q(PART)*Vons(PART) *{ Rocs(POIo)*sin(Aθoc(POIo)) *RDEpdh(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 - 2 *Rpch(POIo(t),t) *{ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t } /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART) *t]^2 }^2 } = (1 - λ(Vons(PART)))*Q(PART)*Vons(PART) *{ Rocs(POIo)*sin(Aθoc(POIo)) *RDEpdh(POIo(t),t) - 2*{ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t } *Rpch(POIo(t),t) } / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART) *t]^2 }^2 Summarizing : (mathH)/* where RDEpdh(POIo(t),t) is anchored at end of Rpch(POIo(t),t) and is at angle Aθpc(POIo(t),t) + PI/2, ie perpendicular to Rpch(POIo(t),t), angle Aφpc(POIo(t),t) doesn't change /% ∂[∂(t): ET_LENZodv(POIo(t),t)] = (1 - λ(Vons(PART)))*Q(PART)*Vons(PART) * { Rocs(POIo)*sin(Aθoc(POIo)) *RDEpdh(POIo(t),t) - 2*{ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t} *Rpch(POIo(t),t) } / {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo)) *Vons(PART)*t + [Vons(PART)*t]^2}^2 (endMath) /*+-----+ LIMIT CHECKS : From "Dimensional analysis (Gaussian units)" in file "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt" : ∂[∂(t): E] units = charge/length^2/time Dimensional consistency - OK, as RHS & LHS reduce to (charge/length^2/time). (just a quick eyeball look...) /%Equation (5) is (1 - λ(Vons(PART))) times "∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)]" Equation (9), as expected. /********************* >>>>>>>>> ∂[∂(t): BT_LENZodv(POIo(t),t)] = ∂[∂(t): BT_LENZpdv(POIo(t),t)] using Lenz's Induction Law, cheating scalar substitutions /%I will simplify the results of "∂[∂(t): BT_LENZodv(POIo(t),t)] = ∂[∂(t): BT_LENZpdv(POIo(t),t)] without use of Lenz's Induction Law" by substituting for EI_LENZods(POIo(t),t) & ∂[∂(t): EI_LENZods(POIo(t),t)] through the use of Lenz's Law. /*+-----+ (RFp) basis /%From "∂[∂(t): BT_LENZodv(POIo(t),t)] = ∂[∂(t): BT_LENZpdv(POIo(t),t)] without use of Lenz's Induction Law" : (6)* ∂[∂(t): BT_LENZpdv(POIo(t),t)] = ∂[∂(t): BT_LENZodv(POIo(t),t)] = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*Q(PART) /Rpcs(POIo(t),t)^3 - EI_LENZpds(POIo(t),t)/Rpcs(POIo(t),t) - ∂[∂(t): EI_LENZpds(POIo(t),t)]/Vons(PART)/cos(Aθpc(POIo(t),t)) } Improper derivative of Rodh(Vonv_X_Rpcv(POIo(t),t)) ] -> should use Rpdv(Vonv_X_Rpcv(POIo(t),t))/|Rpdv(Vonv_X_Rpcv(POIo(t),t))| with Kahan's formulation. (see From "EI_LENZods(POIo(t),t) = EI_LENZpds(POIo(t),t), using Lenz's Induction Law" : (2)* EI_LENZpds(POIo(t),t) = λ(Vons(PART))*|Q(PART)|/Rpcs(POIo(t),t)^2 From "∂[∂(t): EI_LENZods(POIo(t),t)] = ∂[∂(t): EI_LENZpds(POIo(t),t)] - using Lenz's Law, based on proper vector approach" : (2)** ∂[∂(t): EI_LENZods(POIo(t),t)] = 2*λ(Vons(PART))*|Q(PART)|*cos(Aθpc(POIo(t),t))*Vons(PART)/Rpcs(POIo(t),t)^3 Subbing (2)* & (2)** into (6)* : (6)* ∂[∂(t): BT_LENZpdv(POIo(t),t)] = ∂[∂(t): BT_LENZodv(POIo(t),t)] = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*Q(PART) /Rpcs(POIo(t),t)^3 - EI_LENZpds(POIo(t),t)/Rpcs(POIo(t),t) - ∂[∂(t): EI_LENZpds(POIo(t),t)]/Vons(PART)/cos(Aθpc(POIo(t),t)) } = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3 *Q(PART) /Rpcs(POIo(t),t)^3 - λ(Vons(PART))*|Q(PART)| /Rpcs(POIo(t),t)^2 /Rpcs(POIo(t),t) - 2*λ(Vons(PART))*|Q(PART)|*cos(Aθpc(POIo(t),t))*Vons(PART)/Rpcs(POIo(t),t)^3 /Vons(PART)/cos(Aθpc(POIo(t),t)) } = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3 *Q(PART) /Rpcs(POIo(t),t)^3 - λ(Vons(PART))*|Q(PART)|/Rpcs(POIo(t),t)^3 - 2*λ(Vons(PART))*|Q(PART)|/Rpcs(POIo(t),t)^3 } = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *|Q(PART)|/Rpcs(POIo(t),t)^3 *{ 3 - 3*λ(Vons(PART)) } = 3*(1 - λ(Vons(PART)))*|Q(PART)|*Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 *Rodh(Vonv_X_Rpcv(POIo(t),t)) Summarizing : (mathH) ∂[∂(t): BT_LENZpdv(POIo(t),t)] = 3*(1 - λ(Vons(PART)))*|Q(PART)|*Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 *Rodh(Vonv_X_Rpcv(POIo(t),t)) (endMath) /*+--+ Limit CHECKS Dimensional consistency : From file "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt" sub-sub-section "Dimensional analysis (Gaussian units)" : As a convention here take [unit vectors, c] as dimensionless . ∂[∂(t): B] units = charge/length/time^2 /% (2) ∂[∂(t): BT_LENZpdv(POIo(t),t)] = ∂[∂(t): BT_LENZodv(POIo(t),t)] /* (charge/length/time^2) /% = 3*(1 + λ(Vons(PART)))*Q(PART)*Vons(PART)^2/c*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 *Rodh(Vonv_X_Rpcv(POIo(t),t)) = charge *(length/time)^2 /length^3 = (charge/length/time^2) /*OK, as all terms reduce to (dimensionless). Note : I have not yet re-derived "∂[∂(t): BT_LENZodv(POIo(t),t)] = ∂[∂(t): BT_LENZpdv(POIo(t),t)] using Lenz's Induction Law" using a proper vector approach. /*+-----+ (RFo) basis /%From "∂[∂(t): BT_LENZodv(POIo(t),t)] = ∂[∂(t): BT_LENZpdv(POIo(t),t)] without use of Lenz's Induction Law" : (6)* ∂[∂(t): BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t)] = Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t))* { - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(3/2) *[ Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } - EI_LENZods(POIo(t),t) ] + sin(Aθpc(POIo(t),t)) *[ Q(PART)*-2*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 - ∂[∂(t): EI_LENZods(POIo(t),t)] ] } /*Note that BOTH [ EI_LENZodv(POIo(t),t), ∂[∂(t): EI_LENZodv(POIo(t),t)] ] have changing direction & magnitude, and are NOT collinear with the (constant magnitude & direction) unit vector, Rodh(Vonv_X_Rpcv(POIo)). Therefore, I am concerned that direct substitution for [ EI_LENZods(POIo(t),t), ∂[∂(t): EI_LENZods(POIo(t),t)] ] into (6)* will NOT be appropriate, and that a more detailed "vector calculus" check is needed!! However, for now I proceed with direct substitution... /%From "EI_LENZods(POIo(t),t) = EI_LENZpds(POIo(t),t), using Lenz's Induction Law" : (3)* EI_LENZods(POIo(t),t) = EI_LENZpds(POIo(t),t) = λ(Vons(PART))*|Q(PART)| /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } From "∂[∂(t): EI_LENZods(POIo(t),t)] = ∂[∂(t): EI_LENZpds(POIo(t),t)] - using Lenz's Law, based on proper vector approach" : (3)** ∂[∂(t): EI_LENZods(POIo(t),t)] = 2*λ(Vons(PART))*|Q(PART)|*Vons(PART) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^2 Substituting (3)*,(3)** into (6)* : ∂[∂(t): BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t)] = Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t))* { - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(3/2) *[ |Q(PART)| /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } - EI_LENZods(POIo(t),t) ] + sin(Aθpc(POIo(t),t)) *[ |Q(PART)|*-2*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 - ∂[∂(t): EI_LENZods(POIo(t),t)] ] } = Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t))* { - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(3/2) *[ |Q(PART)| /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } - λ(Vons(PART))*|Q(PART)| /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } ] + sin(Aθpc(POIo(t),t)) *[ -2*|Q(PART)|*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 - 2*λ(Vons(PART))*|Q(PART)|*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^2 ] } = Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t))* { - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(3/2) *[ |Q(PART)| /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } * (1 - λ(Vons(PART))) ] + sin(Aθpc(POIo(t),t)) *[ -2*|Q(PART)|*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 * (1 - λ(Vons(PART))) ] } (1) = (1 - λ(Vons(PART)))*|Q(PART)|*Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(3/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } ] + sin(Aθpc(POIo(t),t)) *[ -2*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 ] } from "sin(Aθpc(POIo(t),t))" : (5)* sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (5)* into (1) : (1) ∂[∂(t): BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t)] = (1 - λ(Vons(PART)))*|Q(PART)|*Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(3/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } ] + Rocs(POIo)*sin(Aθoc(POIo)) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) *[ -2*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 ] } = (1 - λ(Vons(PART)))*|Q(PART)|*Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(5/2) ] + Rocs(POIo)*sin(Aθoc(POIo)) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) * -2*Vons(PART) * [ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] ] } = (1 - λ(Vons(PART)))*|Q(PART)|*Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) * [ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) *{ - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) + Rocs(POIo)*sin(Aθoc(POIo))*-2*Vons(PART) } = (1 - λ(Vons(PART)))*|Q(PART)|*Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) * [ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) *{ - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) + - 2*Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) } = (1 - λ(Vons(PART)))*|Q(PART)|*Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) * [ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) * - 3*Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) = 3*(1 - λ(Vons(PART)))*|Q(PART)|*Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) * Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART)* [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) = 3*(1 - λ(Vons(PART)))*|Q(PART)|*Vons(PART)^2/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) * Rocs(POIo)*sin(Aθoc(POIo))* [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) Summarizing : (mathH)/* !!!Note : problem with |Q(PART)| versus Q(PART) ignored here!!! /% BT_LENZodv(POIo(t),t)] = 3*(1 - λ(Vons(PART)))*|Q(PART)|*Vons(PART)^2/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) * Rocs(POIo)*sin(Aθoc(POIo))* [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) (endMath) /*+--+ Limit CHECKS Dimensional consistency : From file "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt" sub-sub-section "Dimensional analysis (Gaussian units)" : As a convention here take [unit vectors, c] as dimensionless . B units = (charge/length/time) (ignoring permittivity & permeability for Gaussian units) ∂[∂(t): B] units = charge/length/time^2 Showing units for (3) : /% ∂[∂(t): BT_LENZpdv(POIo(t),t) charge/length/time^2 = (1 + λ(Vons(PART)))*Q(PART)*Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) (charge) (length/time)^2 *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] [ (length) + (length/time)*time ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 /{ (length)^2 - length*(length/time)*time + [(length/time)*time]^2 }^2 *{ - Rocs(POIo) length /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(1/2) /* /[ (length)^2 - length*(length/time)*time + [(length/time)*time]^2]^(1/2) + -2 } = (charge) * (length/time)^2 * [ (length) + (length/time)*time ] /{ (length)^2 - length*(length/time)*time + [(length/time)*time]^2 }^2 *{ length /[ (length)^2 - length*(length/time)*time + [(length/time)*time]^2]^(1/2) } = charge*(length/time)^2 * (length) /(length)^2^2 * length /(length)^2^(1/2) = charge*(length/time)^2 * (length) /(length)^4 * length /(length) = (charge*length^(4-5)/time^2) = (charge*length/time^2) OK, as all terms reduce to (charge/length/time^2). /*+-----+ Rederiving the (RFp) basis - as a check As is often the case, more compact expressions can result from mixing (RFp) & (RFo) terms. Repeating : /% (3) ∂[∂(t): BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t)] = (1 + λ(Vons(PART)))*Q(PART)*Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 *{ - Rocs(POIo) /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(1/2) + -2 } /*!!!Note : problem with |Q(PART)| versus Q(PART) ignored here!!! /%From "cos(Aθpc(POIo(t),t))" : (1)* cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) From "sin(Aθpc(POIo(t),t))" : (5)* sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (1)* & (5)* into (3) : (3) ∂[∂(t): BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t)] = (1 + λ(Vons(PART)))*Q(PART)*Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *cos(Aθpc(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(2-1/2) *{ - Rocs(POIo) /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(1/2) + -2 } = (1 + λ(Vons(PART)))*Q(PART)*Vons(PART)^2/c*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) *{ - Rocs(POIo)*sin(Aθpc(POIo(t),t)) /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(1/2) + -2*sin(Aθpc(POIo(t),t)) } = (1 + λ(Vons(PART)))*Q(PART)*Vons(PART)^2/c*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) *{ - sin(Aθpc(POIo(t),t)) + -2*sin(Aθpc(POIo(t),t)) } = (1 + λ(Vons(PART)))*Q(PART)*Vons(PART)^2/c*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) * -3*sin(Aθpc(POIo(t),t)) (4) = -3*(1 + λ(Vons(PART)))*Q(PART)*Vons(PART)^2/c *sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) From "Rpcs(POIo(t),t)" : (3) Rpcs(POIo(t),t) = |Rpcv(POIo(t),t)| = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (3)* into (4) ∂[∂(t): BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t)] = -3*(1 + λ(Vons(PART)))*Q(PART)*Vons(PART)^2/c *sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) = -3*(1 + λ(Vons(PART)))*Q(PART)*Vons(PART)^2/c *sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) Summarizing : (5) ∂[∂(t): BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t)] = -3*(1 + λ(Vons(PART)))*Q(PART)*Vons(PART)^2/c*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 *Rodh(Vonv_X_Rpcv(POIo(t),t)) (RFp) basis - comparing to (2) above : (2) ∂[∂(t): BT_LENZpdv(POIo(t),t)] = ∂[∂(t): BT_LENZodv(POIo(t),t)] = 3*(1 + λ(Vons(PART)))*Q(PART)*Vons(PART)^2/c*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 *Rodh(Vonv_X_Rpcv(POIo(t),t)) /*+--+ Limit CHECKS AWESOME! The ONLY difference is the minus sign, but I'll have to better handle vector norms (as mentioned above!). While this "ring around the rosie" verification doesn't prove that either (2) or (3) are correct, it does give confidence at least in the mechanics of the derivation of (3) from (2) (closed loop calcs). /********************************************** >>>>>> Basics and Calculus using Lucas's results from Thomas Barnes iterations /********************* >>>>>>>>> Thomas Barnes iterations, it's context and relation to Lenz's Induction Law Expressions for EOodv(POIo(t),t) and its derivatives can be derived from simple geometry and basic laws. Lenz's law can be applied by using an unknown function of velocity, lambda(v), in order to derive expressions for the INDUCED & TOTAL electric and magnetic fields. This is what I have done in the sub-section of this document "III.4 Basics and Calculus using Lenz's Induction Law". However, the way that I am using terminology here, is that in the section "", the phrase "using Lenz's Induction Law" means that only the "first effect" of the E field is accounted for. In the section "III.5 Basics and Calculus using Lucas's results from Thomas Barnes iterations", an infinite series of feedbacks is accounted for to estimate the full effect of the E field, and therefore to estimate the full (rather than "first pass") magnetic field as well. The "first pass" results in the sub-sub-sections "using Lenz's Induction Law", turns out to yield the simple expression for lambda(v) : (1) EI = lambda(v)*E0 , where lambda(v) = beta^2 = (v/c)^2 beta^2 = (v/c)^2 which is commonly known as beta. Beta is part of the relativistic factor that arises from Relativity Theory, but that is shown to be superfluous and more "grounded" in the Barnes approach. Lucas follows Thomas Barnes work in using an iterative calculation to derive the full expressions for E & B fields, starting with "first pass" estimates arising from "using Lenz's Induction Law". The full ET (total, iterated Electric field or magnetic field), that arises is described by : /% (mathH) ETodv(POIo,t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*E0odv (endMath) where f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/(1 - [β*sin(Aθoc(POIo))]^2)^(3/2) , and again β = Vonv(PART)/c (3) BI = Vonv(PART)/c X ETodv(POIo,t) where "X" denotes the vector cross-product I do the iterative calculation to derive f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) in the file "Howell - math of Lucas Universal Force.ndf", which is the main document that assesses Lucas's Chapter 4. /********************* >>>>>>>>> Howell's correction to the f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) iteration factor ???NO - this would be for ∂[∂(t): EI] !!!??? In the sub-sub-sections below, I have simply used Lucas's final expressions after deriving /% f_BARNES(β,lambda). /*These are compared to my own results from applying Lenz's Indiuction Law, using hte general & unknown lambda(v) function of speed. /********************* >>>>>>>>> ET_BARNodv(POIo(t),t) = ET_BARNpdv(POIo(t),t), using Lucas's results from Thomas Barnes iterations (needs verification!) /*+-----+ (RFp) basis /%NOTE: EI_LENZpdv(POIo(t),t) = EIodv(POIo,t) I am ASSUMING that E0odv(POIo,t) & EIodv(POIo,t) are in the same direction, Rpch(POIo(t),t)!!! /*From Lucas's book p73h0.8 Equation (4-43) : /%(4-43) ET_BARNpdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*E0pdv(POIo(t),t) where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2) β = Vons(PART)/c in the context of Chapter 4 From "E0odv(POIo,t) = E0pdv(POIo(t),t)" : (1)* E0pdv(POIo(t),t) = E0odv(POIo,t) = Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) Subbing (1)* into (4-43) : ET_BARNpdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*E0pdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)/Rpcs(POIp)^2*Rpch(POIo(t),t) (mathH)/* where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2), β = Vons(PART)/c = (λ(Vons(PART)))^(1/2) in the context of Chapter 4 /% ET_BARNpdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)/Rpcs(POIp)^2*Rpch(POIo(t),t) (endMath) /*+--+ CHECKS Dimensional consistency - OK, as all terms reduce to (charge/length^2). (ignoring electric permeability for Gaussian coordinates) Note that "beta = Vons(PART)/c" is dimensionless /*+-----+ (RFo) basis As the E field is a function of the POIo and the charged particle, but independent of the reference frame : /%(1) ET_BARNodv(POIo(t),t) = ET_BARNpdv(POIo(t),t) /*From (1) in "(RFp) basis" above : /% (1)* ET_BARNpdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)/Rpcs(POIp)^2*Rpch(POIo(t),t) where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2) β = Vons(PART)/c = (λ(Vons(PART)))^(1/2) in the context of Chapter 4 From "Rpcs(POIo(t),t)" : (3)* Rpcs(POIo(t),t) = |Rpcv(POIo(t),t)| = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (1)* & (3)* into (1) : ET_BARNodv(POIo(t),t) = ET_BARNpdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)/Rpcs(POIp)^2*Rpch(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Rpch(POIo(t),t) /Rpcs(POIp)^2 = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^2 Leaving [_BARNES(Vonv(PART),Aθpd(POIo(t),t)), Rpch(POIo(t),t)] in (RFp) coordiantes for convenience, and summarizing : (mathH)/* where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2), β = Vons(PART)/c = (λ(Vons(PART)))^(1/2) in the context of Chapter 4 /% ET_BARNodv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Rpch(POIo(t),t) / {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}^(1/2)^2 (endMath) /*+--+ CHECKS Dimensional consistency - OK, as all terms reduce to (charge/length^2). (ignoring electric permeability for Gaussian coordinates) Note that "beta = Vons(PART)/c" is dimensionless, as is the unit vector Rpch(POIo(t),t) /********************* /%>>>>>>>>> ET_BARNods(POIo(t),t) = ET_BARNpds(POIo(t),t), using Lucas's results from Thomas Barnes iterations /*See the file "Howell - math of Lucas Universal Force.ndf", equation (4-33) for the details of the iterations. Here I address the question of the relative DIRECTIONs of E0odv(POIo,t) & EIodv(POIo,t), and rexpress Lucas's result for EIpds(POIo(t),t). NOTE: EIpdv(POIo(t),t) = EIodv(POIo,t) I am ASSUMING that E0odv(POIo,t) & EIodv(POIo,t) are in the same direction!!! Note : Lenz's Law (Lucas p64h0.5 Eq (4-5), p70h0.9 Eq (4-31)) provides assurance that E0odv(POIo,t) & EIodv(POIo,t) are in the same direction ... ??? /*+-----+ Proof that E0odv(POIo,t) & EIodv(POIo,t) are in the same direction This needs to be done later, but at first glance, it seems reasonable to assume that this is the case, as per Lenz's induction law, which is Equation (4-6) p64h0.5 in Lucas's book. Later ..... a more formal proof must be done. /*+-----+ (RFp) basis By definition : /%(1) ET_BARNpds(POIo(t),t) = |ET_BARNpdv(POIo(t),t)| From "ET_BARNodv(POIo(t),t) = ET_BARNpdv(POIo(t),t), using Lucas's results from Thomas Barnes iterations" : (1)* ET_BARNpdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)/Rpcs(POIp)^2*Rpch(POIo(t),t) where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2) β = Vons(PART)/c = (λ(Vons(PART)))^(1/2) in the context of Chapter 4 Subbing (1)* into (1) : ET_BARNpds(POIo(t),t) = |ET_BARNpdv(POIo(t),t)| = |f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)/Rpcs(POIp)^2*Rpch(POIo(t),t)| As [f_BARNES(Vonv(PART),Aθpd(POIo(t),t)),|Q(PART)|,Rpcs(POIp)^2] are >=0, |Rpch(POIo(t),t)| = 1 (unit vector) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|/Rpcs(POIp)^2 Summarizing : (mathH)/* where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2), β = Vons(PART)/c = (λ(Vons(PART)))^(1/2) in the context of Chapter 4 /% ET_BARNpds(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|/Rpcs(POIp)^2 (endMath) /*+--+ CHECKS Dimensional consistency - OK, as all terms reduce to (charge/length^2). (ignoring electric permeability for Gaussian coordinates) Note that [f_BARNES(Vonv(PART),Aθpd(POIo(t),t)), lambda(v), beta] are all dimensionless /*+-----+ (RFo) basis Again, by definition : /%(1) ET_BARNods(POIo(t),t) = |ET_BARNodv(POIo(t),t)| NO LONGER : IMPORTANT!!! Here I have assumed that Lucas's use of sin(Aθod(POIo)) was really intended to be sin(Aθpd(POIo(t),t)), as appears in the result above. From "ET_BARNodv(POIo(t),t) = ET_BARNpdv(POIo(t),t), using Lucas's results from Thomas Barnes iterations" "(RFo) basis" : (2)* ET_BARNodv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2) β = Vons(PART)/c = (λ(Vons(PART)))^(1/2) in the context of Chapter 4 Subbing (2)* into (1) : ET_BARNods(POIo(t),t) = |ET_BARNodv(POIo(t),t)| = | f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } | As [ f_BARNES(Vonv(PART),Aθpd(POIo(t),t)),Rpcs(POIp)^2] are >=0, and |Rpch(POIo(t),t)| = 1 (unit vector) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)| /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } Summarizing : (mathH)/* where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2), β = Vons(PART)/c = (λ(Vons(PART)))^(1/2) in the context of Chapter 4 /% ET_BARNods(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)| /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2} (endMath) /*+--+ CHECKS Dimensional consistency : /%From 4) ET_BARNods(POIo(t),t) = Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } *(1 - β) /{1 - [ β*Rocs(POIo)*sin(Aθoc(POIo)) /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ]^2 }^(3/2) where : (2) β = Vons(PART)/c in the context of Chapter 4 Note that "β = Vons(PART)/c" is dimensionless, and expressing in units : = charge / length^2 *dimensionless /{1 - [ length/length ) ]^2}^(3/2) = charge/length^2 /*OK, as all terms reduce to (charge/length^2). (ignoring electric permeability for Gaussian coordinates) Note that "beta = Vons(PART)/c" is dimensionless /********************* >>>>>>>>> BT_BARNodv(POIo(t),t) = BT_BARNpdv(POIo(t),t), using Lucas's results based on Thomas Barnes iterations (needs verification!) /*+-----+ (RFp) basis As per Lucas p73h0.75 Equation (4-43) part 2 (ie 4-43b or something like that) : /%(4-43b) BI_BARNpdv(POIo(t),t) = Vonv(PART)/c X ET_BARNpdv(POIo(t),t) /*Under the assumption that the are NO other magnetic fields B from magnetic particles or other electric fields : /%(1) BT_BARNpdv(POIo(t),t) = BI_BARNpdv(POIo(t),t) From "ET_BARNodv(POIo(t),t) = ET_BARNpdv(POIo(t),t), using Lucas's results from Thomas Barnes iterations" : (1)* ET_BARNpdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)/Rpcs(POIp)^2*Rpch(POIo(t),t) where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2) β = Vons(PART)/c = (λ(Vons(PART)))^(1/2) in the context of Chapter 4 Subbing (1)* into (1) : BT_BARNpdv(POIo(t),t) = Vonv(PART)/c X ET_BARNpdv(POIo(t),t) (2) = Vonv(PART)/c X f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)/Rpcs(POIp)^2*Rpch(POIo(t),t) Considering the vector cross-product : Here the situation is as described in "BTodv(POIo,t) = BTpdv(POIo(t),t)" : (6) BTpdv(POIo(t),t) = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t) ] where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Aφpd(Vonv_X_Rpcv(POIo(t),t)) & Rodh(Vonv_X_Rpcv(POIo(t),t)) are CONSTANT for Chapter 4 See also numerous warnings and corrections in that sub-sub-section and others... Now BI_BARNpdv(POIo(t),t) IS BTpdv(POIo(t),t) for which f_BARNES provides a solution for EIpds(POIo(t),t) : [ Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t) ] = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)/Rpcs(POIp)^2 So here we can make the SAME statements for the direction (unit vector) for BI_BARNpdv(POIo(t),t) as we did for BTpdv(POIo(t),t). Re-stating (2) : BT_BARNpdv(POIo(t),t) = Vonv(PART)/c X f_BARNES(Vonv(PART),Aθpd(POIo(t),t))* Q(PART) /Rpcs(POIp)^2*Rpch(POIo(t),t) = Vons(PART)/c*sin(Aθpc(POIo(t),t))*f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|/Rpcs(POIp)^2*Rodh(Vonv_X_Rpcv(POIo(t),t)) Summarizing : (mathH)/* where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Aφpd(Vonv_X_Rpcv(POIo(t),t)) & Rodh(Vonv_X_Rpcv(POIo(t),t)) are CONSTANT for /% BTI_BARNpdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)/c*sin(Aθpc(POIo(t),t))/Rpcs(POIp)^2*Rodh(Vonv_X_Rpcv(POIo(t),t)) (endMath) /*+-----+ (RFo) basis Under the assumption that the are NO other magnetic fields B from magnetic particles or other electric fields : /%(1) BT_BARNodv(POIo(t),t) = BI_BARNodv(POIo(t),t) From "ET_BARNodv(POIo(t),t) = ET_BARNpdv(POIo(t),t), using Lucas's results from Thomas Barnes iterations" (RFo) basis : (2)* ET_BARNodv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2) β = Vons(PART)/c = (λ(Vons(PART)))^(1/2) in the context of Chapter 4 Subbing (2)* into (1) : BT_BARNodv(POIo(t),t) = Vonv(PART)/c X ET_BARNodv(POIo(t),t) (2) = Vonv(PART)/c X f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } Notice that we retain "Aθpd(POIo(t),t))" (RFp format) for convenience, but in any case it is a factor for f_BARNES, as opposed to being directly in the equation (2). Considering the vector cross-product : As with the RFo basis and as described in "BTodv(POIo,t) = BTpdv(POIo(t),t)" : (9) BTodv(POIo,t) = Vons(PART)/c *Rocs(POIo)*sin(Aθoc(POIo)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) - EIods(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ] where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) See also numerous warnings and corrections in that sub-sub-section and others... Now BI_BARNodv(POIo(t),t) IS BTodv(POIo,t) for which f_BARNES provides a solution for EIpds(POIo(t),t) : So here we can make the SAME statements for the direction (unit vector) for BI_BARNpdv(POIo(t),t) as we did for BTpdv(POIo(t),t). Re-stating (2), and retaining (RFp) terms [Rodh(Vonv_X_Rpcv(POIo(t),t)), sin(Aθpc(POIo(t),t))] for great convenience : BT_BARNodv(POIo(t),t) = Vonv(PART)/c X f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } = Vons(PART)/c*sin(Aθpc(POIo(t),t))*f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Rodh(Vonv_X_Rpcv(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } (3) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } From "sin(Aθpc(POIo(t),t))" : (5)* sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (5)* into (3) : BT_BARNodv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)/c *Rodh(Vonv_X_Rpcv(POIo(t),t)) *sin(Aθpc(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)/c *Rodh(Vonv_X_Rpcv(POIo(t),t)) * Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)/c*Rocs(POIo)*sin(Aθoc(POIo)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) Summarizing : (mathH)/* where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Aφpd(Vonv_X_Rpcv(POIo(t),t)) & Rodh(Vonv_X_Rpcv(POIo(t),t)) are CONSTANT for /% BT_BARNodv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)/c*Rocs(POIo)*sin(Aθoc(POIo)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}^(3/2) (endMath) /********************* /%>>>>>>>>> ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))], using Lucas's results based on Thomas Barnes iterations (needs verification!) /*+-----+ (RFp) FORMAT From Lucas's book p73h0.8 Equation (4-43), and my use of the symbol f_BARNES : /%(4-43) ET_BARNpdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*E0pdv(POIo(t),t) where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2) β = Vons(PART)/c in the context of Chapter 4 Therefore, by definition in (4-43) : (1) f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2) Note that f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) is a scalar (+ve,-ve), not a vector. With normal assumptions, f_BARNES >= 0, but I don't enforce that here. Taking the derivative of (1) : ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] = ∂[∂(t): (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2)] Given that [Vonv(PART), β] are constant in Chapter 4 : = (1 - β^2)*∂[∂(t): {1 - [β*sin(Aθpd(POIo(t),t))]^2}^( - 3/2)] = (1 - β^2)*(-3/2) *{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(-5/2) *∂[∂(t): 1 - [β*sin(Aθpd(POIo(t),t))]^2] = (1 - β^2)*(-3/2) *{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(-5/2) *(-1) *∂[∂(t): [β*sin(Aθpd(POIo(t),t))]^2] = (1 - β^2)*(-3/2) *{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(-5/2) *(-1)*2*β*sin(Aθpd(POIo(t),t)) *∂[∂(t): β*sin(Aθpd(POIo(t),t))] = (1 - β^2)*(-3/2) *{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(-5/2) *(-1)*2*β*sin(Aθpd(POIo(t),t))*β*∂[∂(t): sin(Aθpd(POIo(t),t))] (2) = 3*β^2*(1 - β^2)*{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(-5/2)*sin(Aθpd(POIo(t),t)) *∂[∂(t): sin(Aθpd(POIo(t),t))] From "∂[∂(t): sin(Aθpc(POIo(t),t))]" RFp basis : (2)* ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) Subbing (2)* into (2) : (2) ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] = 3*β^2*(1 - β^2)*{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(-5/2)*sin(Aθpd(POIo(t),t)) *∂[∂(t): sin(Aθpd(POIo(t),t))] = 3*β^2*(1 - β^2)*{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(-5/2)*sin(Aθpd(POIo(t),t)) *Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) (3) = 3*β^2*(1 - β^2)*{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(-5/2) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) Summarizing : (mathH) ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) (endMath) /*+-----+ (RFo) basis Converting RFp (3) above to RFo terms : (3)* ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] = 3*beta^2*(1 - beta^2)*{1 - [beta*sin(Aθpd(POIo(t),t))]^2}^(-5/2) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) From "sin(Aθpc(POIo(t),t))" RFo basis : (5)* sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) From "cos(Aθpc(POIo(t),t))" RFo basis: (1)* cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) From "Rpcs(POIo(t),t)" RFo basis : (3)** Rpcs(POIo(t),t) = |Rpcv(POIo(t),t)| = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (5)*,(1)*,(3)** into (3)* (except for {1 - [beta*sin(Aθpd(POIo(t),t))]^2} term) : ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] = 3*beta^2*(1 - beta^2)*{1 - [beta*sin(Aθpd(POIo(t),t))]^2}^(-5/2) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) = 3*beta^2*(1 - beta^2)*{1 - [beta*sin(Aθpd(POIo(t),t))]^2}^(-5/2) *Vons(PART) *sin(Aθpc(POIo(t),t))^2 *cos(Aθpc(POIo(t),t)) /Rpcs(POIo(t),t) = 3*beta^2*(1 - beta^2)*{1 - [beta*sin(Aθpd(POIo(t),t))]^2}^(-5/2) *Vons(PART) *Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) (1) = 3*beta^2*(1 - beta^2)*{1 - [beta*sin(Aθpd(POIo(t),t))]^2}^(-5/2) *Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) Subbing for f_BARNES : = 3*beta^2/{1 - [beta*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) Summarizing : (mathH) ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] = 3*beta^2/{1 - [beta*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) *[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}^(3/2) (endMath) /********************* /%>>>>>>>>> ∂[∂(t): ET_BARNodv(POIo(t),t)] = ∂[∂(t): ET_BARNpdv(POIo(t),t)], using Lucas's results based on Thomas Barnes iterations (needs verification!) /*+-----+ (RFp) FORMAT From Lucas's book p73h0.8 Equation (4-43) : /%(4-43) ET_BARNpdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*E0pdv(POIo(t),t) where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2) β = Vons(PART)/c in the context of Chapter 4 From first principles : ∂[∂(t): ET_BARNpdv(POIo(t),t)] = ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*E0pdv(POIo(t),t)] (1) = ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))]*E0pdv(POIo(t),t) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*∂[∂(t): E0pdv(POIo(t),t)] /*+-----+ FIRST TERM - /%(2) ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))]*E0pdv(POIo(t),t) From "∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))], using Lucas's results based on Thomas Barnes iterations" : (4)* ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) From "E0odv(POIo,t) = E0pdv(POIo(t),t)" : (1)* E0pdv(POIo(t),t) = E0odv(POIo,t) = Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) Subbing (4)* & (1)* into (2) : ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))]*E0pdv(POIo(t),t) = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) *Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) (3) = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Q(PART)*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3*Rpch(POIo(t),t) /*+--+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (charge/length^2/time). (ignoring electric permeability for Gaussian coordinates) /% (3) ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))]*E0pdv(POIo(t),t) 1/time (charge/length^2) = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) dimensionless *Q(PART)*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3*Rpch(POIo(t),t) charge (length/time) /length^3 = charge/length^2/time /*+-----+ SECOND TERM - /%(4) f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*∂[∂(t): E0pdv(POIo(t),t)] From "∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)]" RFp basis : (6)* ∂[∂(t): E0pdv(POIo(t),t)] = Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[ sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) ] where : Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 Subbing (6)* into (4) : (5) f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*∂[∂(t): E0pdv(POIo(t),t)] = f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[ sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) ] where : Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 /*+--+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (charge/length^2/time). (ignoring electric permeability for Gaussian coordinates) /%(5) f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*∂[∂(t): E0pdv(POIo(t),t)] ∂[∂(t): E] units = charge/length^2/time = f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 charge * (length/time) /length^3 *[ sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) ] (no units this line) = (charge/length^2/time) /*+-----+ COMBINING TERMS Subbing (3) & (5) into (1) : /% (1) ∂[∂(t): ET_BARNpdv(POIo(t),t)] = ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] *E0pdv(POIo(t),t) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *∂[∂(t): E0pdv(POIo(t),t)] = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Q(PART)*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3*Rpch(POIo(t),t) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[ sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) ] = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 { 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} *sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) + *[ sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) ] } = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[ Rpch(POIo(t),t) *{ 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t)) + 2*cos(Aθpc(POIo(t),t)) } + RDEpdh(POIo(t),∂(t)) *sin(Aθpc(POIo(t),t)) ] = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[ Rpch(POIo(t),t) *cos(Aθpc(POIo(t),t))*{ 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*sin(Aθpc(POIo(t),t))^2 + 2 } + RDEpdh(POIo(t),∂(t)) *sin(Aθpc(POIo(t),t)) ] Summarizing : (mathH)/* where : Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 /% ∂[∂(t): ET_BARNpdv(POIo(t),t)] = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[ Rpch(POIo(t),t) *cos(Aθpc(POIo(t),t))*{ 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*sin(Aθpc(POIo(t),t))^2 + 2 } + RDEpdh(POIo(t),∂(t)) *sin(Aθpc(POIo(t),t)) ] (endMath) 01Apr2016 - I don't trust this at all! /*+--+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (charge/length^2/time). (ignoring electric permeability for Gaussian coordinates) (same as for second term...) /*+-----+ SPECIAL CHECK As per the note above "This problematic in this derivation!!!!", this result must be checked to see if the original assumption regarding the use of differentiating ∂[∂(t): ET_BARNpds(POIo(t),t)]. Later ... /*+-----+ (RFo) basis : Following the same appraoch as for the RFp basis above From first principles : /% ∂[∂(t): ET_BARNodv(POIo(t),t)] = ∂[∂(t): ET_BARNpdv(POIo(t),t)] = ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*E0odv(POIo,t)] (1) = ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))]*E0odv(POIo,t) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*∂[∂(t): E0odv(POIo,t)] /*+-----+ FIRST TERM - /%(2) ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))]*E0odv(POIo,t) From "∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))], using Lucas's results based on Thomas Barnes iterations" : (2)* ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) From "E0odv(POIo,t) = E0pdv(POIo(t),t)" : (5)* E0odv(POIo) = E0pdv(POIo) = Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } where Rpch(POIo(t),t) = Rodh(POIo) = displacement vector [start : (POIo), length : 1, theta : arccos(Aθpc(POIo(t),t)), phi : Aφoc(POIo)] and cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) Subbing (2)* & (5)* into (2) : (2) ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))]*E0odv(POIo,t) = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) *Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } (3) = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Q(PART)*Rpch(POIo(t),t)*Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) /*+-----+ SECOND TERM - /%(4) f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*∂[∂(t): E0odv(POIo,t)] From "∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)]" RFp basis : (9) ∂[∂(t): E0pdv(POIo(t),t)] = Q(PART)*Vons(PART) *[ Rocs(POIo)*sin(Aθoc(POIo)) *RDEpdh(POIo(t),∂(t)) + 2*[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rpch(POIo(t),t) ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 where : Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 Aθpc(POIo(t),t) = arcsin [ Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ] = arccos [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) ] Subbing (9)* into (4) (5) f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*∂[∂(t): E0odv(POIo,t)] = f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Q(PART)*Vons(PART) *[ Rocs(POIo)*sin(Aθoc(POIo)) *RDEpdh(POIo(t),∂(t)) + 2*[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rpch(POIo(t),t) ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 /*+-----+ COMBINING TERMS Subbing (3) & (5) into (1) : (1) ∂[∂(t): ET_BARNodv(POIo(t),t)] = ∂[∂(t): ET_BARNpdv(POIo(t),t)] = ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))]*E0odv(POIo,t) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*∂[∂(t): E0odv(POIo,t)] = 3*beta^2/{1 - [beta*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Q(PART)*Rpch(POIo(t),t)*Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Q(PART)*Vons(PART) *[ Rocs(POIo)*sin(Aθoc(POIo)) *RDEpdh(POIo(t),dt) + 2*[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rpch(POIo(t),t) ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 = Q(PART)*Vons(PART) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 *{ 3*beta^2/{1 - [beta*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Rpch(POIo(t),t)*Rocs(POIo)*sin(Aθoc(POIo)) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *[ Rocs(POIo)*sin(Aθoc(POIo)) *RDEpdh(POIo(t),dt) + 2*[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rpch(POIo(t),t) ] } = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Vons(PART) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 *{ Rpch(POIo(t),t) *[ 3*beta^2/{1 - [beta*sin(Aθpd(POIo(t),t))]^2} *Rocs(POIo)*sin(Aθoc(POIo)) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) + 2*[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] ] + RDEpdh(POIo(t),dt) *Rocs(POIo)*sin(Aθoc(POIo)) } = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Vons(PART) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 *{ Rpch(POIo(t),t) *[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *[ 3*beta^2/{1 - [beta*sin(Aθpd(POIo(t),t))]^2} * Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) + 2 ] + RDEpdh(POIo(t),dt) *Rocs(POIo)*sin(Aθoc(POIo)) } Summarizing : (mathH)/* where : Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),dt) is at angle Aθpd(RDEpdh(POIo(t),dt)) = Aθpc(POIo(t),t) + PI/2 /% ∂[∂(t): ET_BARNodv(POIo(t),t)] = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Vons(PART) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 *{ Rpch(POIo(t),t) *[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *[ 3*beta^2/{1 - [beta*sin(Aθpd(POIo(t),t))]^2} * Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) + 2 ] + RDEpdh(POIo(t),dt) *Rocs(POIo)*sin(Aθoc(POIo)) } (endMath) /*+--+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (charge/length^2/time). (ignoring electric permeability for Gaussian coordinates) /%(6) ∂[∂(t): ET_BARNodv(POIo(t),t)] (charge/length^2/time) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Vons(PART) charge * length/time / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 /length^4 *{ Rpch(POIo(t),t) *[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] length *[ 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} * Rocs(POIo)*sin(Aθoc(POIo)) length /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /length + 2 ] + RDEpdh(POIo(t),∂(t)) *Rocs(POIo)*sin(Aθoc(POIo)) length } /* = charge * length/time /length^4 *length = (charge/length^2/time) /********************* /%>>>>>>>>> ∂[∂(t): BTodv(POIo,t)] = ∂[∂(t): BTpdv(POIo(t),t)], using Lucas's results from Thomas Barnes iterations (needs verification!) Note that for any specific POIo, BTodv(POIo,t) = BTpdv(POIo(t),t) changes in magnitude and in [+,-] direction, but it DOES NOT change in direction of unit vector Rodh(Vonv_X_Rpcv(POIo(t),t)). [Aφpd(POIo(t),t), APod(POIo)] => direction of unit vector Rodh(Vonv_X_Rpcv(POIo(t),t)). where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Aφpd(Vonv_X_Rpcv(POIo(t),t)) & Rodh(Vonv_X_Rpcv(POIo(t),t)) are CONSTANTS for a given POIo Therefore, ∂[∂(t): BTodv(POIo,t)] = ∂[∂(t): BTpdv(POIo(t),t)] is always in direction of the unit vector Rodh(Vonv_X_Rpcv(POIo(t),t)) for a given POIo. It's magnitude channges, but not the direction (except [+,-]) of the unit vector Rodh(Vonv_X_Rpcv(POIo(t),t)). /*+-----+ (RFp) FORMAT /%From "BT_BARNodv(POIo(t),t) = BT_BARNpdv(POIo(t),t), using Lucas's results based on Thomas Barnes iterations" RFp basis : (3)* BT_BARNpdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)/c*sin(Aθpc(POIo(t),t))/Rpcs(POIp)^2*Rodh(Vonv_X_Rpcv(POIo(t),t)) where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Aφpd(Vonv_X_Rpcv(POIo(t),t)) & Rodh(Vonv_X_Rpcv(POIo(t),t)) are CONSTANT for Taking the derivative of (3)* : ∂[∂(t): BT_BARNpdv(POIo(t),t)] = ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)/c*sin(Aθpc(POIo(t),t))/Rpcs(POIp)^2*Rodh(Vonv_X_Rpcv(POIo(t),t))] As [|Q(PART)|,Vons(PART),c,Rodh(Vonv_X_Rpcv(POIo(t),t))] are constant with time (Rodh(Vonv_X_Rpcv(POIo(t),t)) always in same direction!!!) : = |Q(PART)|*Vons(PART)/c *Rodh(Vonv_X_Rpcv(POIo(t),t)) *∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*sin(Aθpc(POIo(t),t))/Rpcs(POIp)^2] (1) = |Q(PART)|*Vons(PART)/c *Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))]* sin(Aθpc(POIo(t),t)) *Rpcs(POIp)^(-2) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *∂[∂(t): sin(Aθpc(POIo(t),t))] *Rpcs(POIp)^(-2) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) * sin(Aθpc(POIo(t),t)) *∂[∂(t): Rpcs(POIp)^( - 2)] } From "∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))], using Lucas's results based on Thomas Barnes iterations" RFp basis : (4)* ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) From "∂[∂(t): sin(Aθpc(POIo(t),t))]" RFp basis : (2)* ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) From "∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|]" RFp basis : (1)* ∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|] = -Vons(PART)*cos(Aθpc(POIo(t),t)) Subbing (4)*,(2)*,(1)* into (1) : (1) ∂[∂(t): BT_BARNpdv(POIo(t),t)] = |Q(PART)|*Vons(PART)/c *Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))]* sin(Aθpc(POIo(t),t)) *Rpcs(POIp)^(-2) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *∂[∂(t): sin(Aθpc(POIo(t),t))] *Rpcs(POIp)^(-2) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) * sin(Aθpc(POIo(t),t)) *∂[∂(t): Rpcs(POIp)^( - 2)] } = |Q(PART)|*Vons(PART)/c *Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} *f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) *sin(Aθpc(POIo(t),t)) *Rpcs(POIp)^(-2) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Rpcs(POIp)^(-2) *Vons(PART)*sin(Aθpc(POIo(t),t)) *cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) * sin(Aθpc(POIo(t),t)) *(-2)*Rpcs(POIp)^(-3)*∂[∂(t): Rpcs(POIp)] } = |Q(PART)|*Vons(PART)/c *Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} *f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) *sin(Aθpc(POIo(t),t)) *Rpcs(POIp)^(-2) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Rpcs(POIp)^(-2) *Vons(PART)*sin(Aθpc(POIo(t),t)) *cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) * sin(Aθpc(POIo(t),t)) *(-2)*Rpcs(POIp)^(-3) *-Vons(PART)*cos(Aθpc(POIo(t),t)) } = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)^2/c*cos(Aθpc(POIo(t),t)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} *sin(Aθpc(POIo(t),t))^2/Rpcs(POIo(t),t) *sin(Aθpc(POIo(t),t)*Rpcs(POIp)^( - 2) + sin(Aθpc(POIo(t),t)) /Rpcs(POIo(t),t) *Rpcs(POIp)^(-2) + sin(Aθpc(POIo(t),t)) *(-2)*Rpcs(POIp)^(-3) *(-1) } = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)^2/c*cos(Aθpc(POIo(t),t)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ { 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} *sin(Aθpc(POIo(t),t))^3/Rpcs(POIo(t),t)^3 + sin(Aθpc(POIo(t),t)) /Rpcs(POIo(t),t)^3 + 2*sin(Aθpc(POIo(t),t)) /Rpcs(POIp)^3 } = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIp)^3 *Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ { 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} *sin(Aθpc(POIo(t),t))^2 + 1 + 2 } = 3*f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIp)^3 *{ { [β*sin(Aθpc(POIo(t),t))]^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} + 1 } *Rodh(Vonv_X_Rpcv(POIo(t),t)) Summarizing : (mathH)/* where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Aφpd(Vonv_X_Rpcv(POIo(t),t)) & Rodh(Vonv_X_Rpcv(POIo(t),t)) are CONSTANT for /% ∂[∂(t): BT_BARNpdv(POIo(t),t)] = 3*f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIp)^3 *{ { [β*sin(Aθpc(POIo(t),t))]^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} + 1} *Rodh(Vonv_X_Rpcv(POIo(t),t)) (endMath) /*+--+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (charge/length/time^2). (ignoring electric permeability for Gaussian coordinates) See file : "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt" sub-section : "Dimensional analysis (Gaussian units)" : /%(3) ∂[∂(t): BT_BARNodv(POIo(t),t)] ∂[∂(t): B] units = charge/length/time^2 = 3*f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIp)^3 charge (length/time)^2 /length^3 *{ { [β*sin(Aθpc(POIo(t),t))]^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} + 1 } *Rodh(Vonv_X_Rpcv(POIo(t),t)) dimensionless line ... /* = charge/length/time^2 /*+-----+ (RFo) basis : /%From "BT_BARNodv(POIo(t),t) = BT_BARNpdv(POIo(t),t), using Lucas's results based on Thomas Barnes iterations" RFo basis : (4)* BT_BARNodv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)/c*Rocs(POIo)*sin(Aθoc(POIo)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Aφpd(Vonv_X_Rpcv(POIo(t),t)) & Rodh(Vonv_X_Rpcv(POIo(t),t)) are CONSTANT for Taking the time derivative of (4)* : ∂[∂(t): BT_BARNodv(POIo(t),t)] = ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)/c*Rocs(POIo)*sin(Aθoc(POIo))*Rodh(Vonv_X_Rpcv(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) ] As [|Q(PART)|,Vons(PART),c,Rocs(POIo),sin(Aθoc(POIo)),cos(Aθoc(POIo)),Rodh(Vonv_X_Rpcv(POIo(t),t))] are constant at a given POIo : = |Q(PART)|*Vons(PART)/c*Rocs(POIo)*sin(Aθoc(POIo)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-3/2) ] (1) = |Q(PART)|*Vons(PART)/c*Rocs(POIo)*sin(Aθoc(POIo)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] *{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-3/2) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *∂[∂(t): {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}^( - 3/2)] } From "∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))], using Lucas's results based on Thomas Barnes iterations" RFo basis : (2)* ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) Looking at : ∂[∂(t): {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}^( - 3/2)] As [Rocs(POIo),sin(Aθoc(POIo)),cos(Aθoc(POIo)),Rodh(Vonv_X_Rpcv(POIo(t),t))] are constant at a given POIo : = ∂[∂(t): {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}^( - 3/2)] = (-3/2) *{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-5/2) ∂[∂(t): {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}] = (-3/2) *{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-5/2) *{ ∂[∂(t): Rocs(POIo)^2] - ∂[∂(t): 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + ∂[∂(t): [Vons(PART)*t]^2] } = (-3/2) *{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-5/2) *{ 0 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART) + 2*Vons(PART)*t*∂[∂(t): Vons(PART)*t] } = (-3/2) *{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-5/2) *{ - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART) + 2*Vons(PART)*t*Vons(PART) } = (-3/2)*2*Vons(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) *{ - Rocs(POIo)*cos(Aθoc(POIo)) + t*Vons(PART) } = 3*Vons(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) *{ + Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t } (2) = 3*Vons(PART) *{ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t } /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) Subbing (2)*,(2) into (1) : ∂[∂(t): BT_BARNodv(POIo(t),t)] = |Q(PART)|*Vons(PART)/c*Rocs(POIo)*sin(Aθoc(POIo)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) *{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-3/2) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) * 3*Vons(PART) *{ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t } /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) } = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)^2/c*Rocs(POIo)*sin(Aθoc(POIo)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) *{ 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} *Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) + 3 } Summarizing : (mathH)/* where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Aφpd(Vonv_X_Rpcv(POIo(t),t)) & Rodh(Vonv_X_Rpcv(POIo(t),t)) are CONSTANT for /% ∂[∂(t): BT_BARNodv(POIo(t),t)] = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)^2/c*Rocs(POIo)*sin(Aθoc(POIo)) * [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rodh(Vonv_X_Rpcv(POIo(t),t)) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) *{ 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} *Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) + 3 } (endMath) /*+--+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (charge/length/time^2). (ignoring electric permeability for Gaussian coordinates) See file : "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt" sub-section : "Dimensional analysis (Gaussian units) " : /%(3) ∂[∂(t): BT_BARNodv(POIo(t),t)] ∂[∂(t): B] units = charge/length/time^2 = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)^2/c*Rocs(POIo)*sin(Aθoc(POIo)) charge (length/time)^2 length * [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rodh(Vonv_X_Rpcv(POIo(t),t)) length / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) /length^5 *{ 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} *Rocs(POIo)*sin(Aθoc(POIo)) length /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) length + 3 } /* = charge * (length/time)^2 * length * length / length^5 = charge/length/time^2 endchapter **************************************** >>> IV. Expressions for the t=0 reference frame (RFt) In order to force myself to be "more correct and consistent" in switching "contexts or reference frames" in going between integrations and derivatives, I found it helpful to define new symbols and the "t=o reference frame" (RFt). 14Jun2016 As I have just adopted this approach, this chapter is woefully inadequate, and in need of much greater detail. Right now, the ONLY content are descriptions of : reference frame switches (RFo -> RFt) resulting from , and RFt -> RFo) and corresponding variable changes only one derivative as shown below. This needs a figure. /********************* >>>>>>>>> Procedures for consistent reference frame switching From below : Chapter "Derivatives & Integrals adapted to Chapter 4", subsection "Summary" Procedures - integrals /* For derivatives in next step : Actually, by incorporating this into the derivatives & integrals, most cases are handled "automatically" /* integration ∫{dAOtc, 0 to Aθpc(POIo(t)=0) : expressions with [Rocs(POIo),Rpcs(POIo(t)=0),AOtc(RFo ),AOtc(RFp) ,E0ods(POIo(t)=0)] } /* result gives [Rocs(POIo),Rpcs(POIo(t),t) ,Aθoc(POIo),Aθpc(POIo(t)=0),E0ods(POIo,t) ] } Procedures - derivatives /* For integrals in next step : Actually, by incorporating this into the derivatives & integrals, most cases are handled "automatically" /* derivative ∂[∂(t): expressions with [Rocs(POIo),Rpcs(POIo(t),t) ,Aθoc(POIo),Aθpc(POIo(t),t),E0ods(POIo,t) ] /* result gives [Rocs(POIo),Rpcs(POIo(t)=0),AOtc(RFo) ,AOtc(RFp) ,E0ods(POIo(t)=0)] /********************* >>>>>>>>> Calculus of RFt /********************* /%>>>>>>>>> ∂[∂(Aθpc): Rpcs(POIo(t),t=0)] = ∂[∂(Aθpc): |Rpcv(POIo(t),t)|] (is this wrong?) /*NOTE : This should be done by a proper vector derivative approach!! 14Jun2016 - Is this WRONG??? CONTEXT : Important -> In the equations of Chapter, we are INTEGRATING with respect to (wrt) Aθpc, rather than differentiating. Simple non-vector approach : /%From "∂[∂(t): Rpcv(POIo(t),t)]" 1*) ∂[∂(t): Rpcv(POIo(t),t)] = -Vonv(PART) From Figure "∂[∂(t): Rpcs(POIo(t),t)]" ROPI2pds(POIo(t),t) = ROPI2ods(POIo) = constant = Rocs(POIo) *sin(Aθoc(POIo)) = Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t)) Or : (1) Rpcs(POIo(t),t) = Rocs(POIo)*sin(Aθoc(POIo))/sin(Aθpc(POIo(t),t)) Differentiating with respect to (wrt) Aθpc : ∂[∂(Aθpc): Rpcs(POIo(t),t)] = ∂[∂(Aθpc): Rocs(POIo)*sin(Aθoc(POIo))/sin(Aθpc(POIo(t),t))] But Rocs(POIo),sin(Aθoc(POIo)) are constants : = Rocs(POIo)*sin(Aθoc(POIo))*∂[∂(Aθpc): sin(Aθpc(POIo(t),t)^( - 1)] = Rocs(POIo)*sin(Aθoc(POIo))*(-1)*sin(Aθpc(POIo(t),t))^(-2)*∂[∂(Aθpc): sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo))*(-1)*sin(Aθpc(POIo(t),t))^(-2)*cos(Aθpc(POIo(t),t) Summarizing : (mathH) ∂[∂(Aθpc): Rpcs(POIo(t),t)] = (-1)*Rocs(POIo)*sin(Aθoc(POIo))*sin(Aθpc(POIo(t),t))^(-2)*cos(Aθpc(POIo(t),t) (endMath) /************************************************* >>> APPENDICES /************************************************* >>>>>> Future extensions of the Universal Force /*/*$ cat >>"$p_augmented" "$d_Lucas""context/future extensions.txt" Here is a [random, scattered, incomplete] list of extensions to Lucas's theory that I am interested in : - relativistic correction factor is not correct, get rid of (1 - beta^2) terms (Ed Dowdye Jr?) - time lags and causality, Oleg Jefimenko - fractional order calculus - ?Johan Suyken's student, now prof? - Comparisons to : - Randall Mills /*_endCmd /********************************************** ; >>>>>>>>> Gaussian versus SI units ; /*/*$ cat >>"$p_augmented" "$d_Lucas""math nomenclature/Gaussian versus SI units.txt" Gaussian versus SI units : - I have NOT properly adjusted formulae from various sources for differences in the units used, whether Gaussian, SI, or other. - This creates some confusion here and there in my review comments. Jackson 1999 p782h0.15 Table 3 provides conversions ?????? put table here!!! /*_endCmd /********************** >>>>>> Symbol checking and translation - short description /*/*$ cat >>"$p_augmented" "$d_Lucas""context/symbols [check, translate].txt" Normal physics symbols and conventions were NOT used in my analysis, as I had trouble ensuring consistency and specificity. Instead, a series of evolving symbol systems were used over the course of the project, as the reader can see from the various files. A short description of symbols, plus links to more detailed programming code that specifies the Also important was the [development, usage] of "Howell's FlatLiner Notation" (HFLN), which is briefly describe in a section below. +-----+ Short description : mathB = "???math Howell YYMMDD HHhMMm.txt???" mathH = "math Howell YYMMDD HHhMMm.txt" mathL = "math Lucas, cos - 1 [yes,no] YYMMDD HHhMMm.txt" where "YYMMDD HHhMMm" = year month day hour minute of the document version For a more complete description, see : "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt" 20Aug2019 - this is out of date, as are symbol translations in this file!! : insert symTrans_lineType '/[#$%^&] ' symbols at the start of lines or within text in a line as needed ( do not include the apostrophes) : '/* ' comment only - no translation or extraction of symbols '/$ ' applies tranList_Lucas_bads to a string '/% ' applies tranList_HFLN_bads to a string '/^ ' applies tranList_Lucas_to_HFLN to a string, translated string has "%^" at start of line to help subsequent editing `& applies tranList_HFLN_to_Lucas to a string, translated string has "$&" at start of line to help subsequent editing symTrans_lineTypes are PERSISTENT for multiple lines - until a symTrans_lineType appears as the first character on a subsequent line. where mixed (cart [Lucas,HFLN] [symbols,expressions]) are in a line - put a `^ as the first character of a line to be translated to HFLN [symbols,expressions] - put a `& as the first character of a line to be translated to Lucas [symbols,expressions] special codes #add_eqn - start of summary descriptions and results for a formula from Lucas's book HFLN - re-expressed equations in HFLN (Howell's Flat-Liner Notation) +-----+ Lists of symbols [valid, invalid] symbols and their translations are provided in : "http://www.BillHowell.ca/Projects - big/Lucas - Universal Force/symbol tables/" 24Oct2019 ... I have yet to post the project to my site !?!? ... tranParn Luca.txt symList good HFLN bad.txt symList good HFLN trn.txt tranExpr Luca.txt tranParn Luca notes.txt tranParn HFLN.txt symList good HFLN.txt tranExpr HFLN.txt symList equation numbers.txt tranParn HFLN notes.txt symList good Luca.txt This needs more explanation ion the future ... +-----+ Prgramming code A detailed description of the symbols and how they were [checked, translated] is available in a collection of programming files : "http://www.BillHowell.ca/bash scripts/" : symbols count in file.sh symbols compliance POI.sh symbols compliance special.sh symbols classify manual.sh symbols check POI.sh "http://www.BillHowell.ca/Qnial/MY_NDFS/" : symbols fix instructions.txt symbols system loaddefs.ndf symbols sort-archive-move, post-manual-class.ndf symbols fix.ndf symbols count in file.sh.txt symbols log.txt symbols extract.ndf symbols translate.ndf symbols compliance special.sh.txt symbols check POI.sh.txt symbols classify manual.sh.txt symbols compliance POI.sh.txt symbols zProcess notes.txt /*_endCmd /******************************************** >>>>>>>>> HFLN = Howells FlatLiner Notation !!!!!!!!!!!!!! 31May2016 /*/*$ cat >>"$p_augmented" "$d_Lucas""context/Howells flat-line notation short description.txt" All material herein follows "Howell's 'FlatLine' formatting conventions" as specified in my document : "$d_Lucas""document/math Variables, notations, styles YYMMDD HHhMMm.txt" This is important, as my notations helped me to be more clear on the concepts and contexts being used. This was a lot of work and headaches, but it did result in a greater ease of writing the derivations, and often the notation made conceptual errors much more visible and trackable. An obvious DISADVANTAGE of my notations is that it will be unfamiliar to others, and considerable time would be required for a reader to become functionally familiar with the notation. . However, it is (theoretically) relatively easy (if practically long and tedious) to program a conversion of the final results of each sub-sub-section from my notation to the standards used by Bill Lucas and the physics community, for example with a preprocessor to Tex or something like that. Rather than re-editing this entire file to use my "most up-to-date" notations, I have simply added the HFLN notations to my comments for SOME of the expressions. Problems have not been fully resolved!! WARNING : Because I rapidly added the expressions for HFLN, many mistakes remain. Being precise about the context for each variable has been a persistent problem. /*_endCmd /********************** >>>>>> Document build short description /*/*$ cat >>"$p_augmented" "$d_Lucas""context/document build short description.txt" +-----+ # This file is created by : - edit "$d_bin""Lucas - formulae & augment.sh" to select one of the "skeleton" documents to process - this file, plus the scripts that it calls, sets the environmental variables - run $ bash "$d_bin""Lucas - formulae & augment.sh" - this creates the "augmented" txtDoc, with [Table of Contents, List of Equations, file insertions, etc] +--+ Normally, you should NOT directly edit txtDocs that have been built from "skeletons". Instead, edit the files that are inserted. Otherwise , files can become inconsistent, and work may be lost. +--+ "/*$" and "/*/*$" The character sequence "/*$" at the start of a line denotes a [Linux, bash, other] command to be executed at that point when the txtDoc is being processed. Common examples are : - insert the date : /*$ echo "version= $date_ymdhm, cos - 1 inclusion : $cos_inclusion" >>"$p_augmented" - insert the name of the txtDoc that is beiong produced : /*$ echo "$p_augmented" >>"$p_augmented" - insert text from a file : /*$ cat >>"$p_augmented" "$d_Lucas""context/text editor - how to set up.txt" Because general commands and special scripts can easily be used, this allows a great deal of [capabilities, flexibilty] to txtDoc processing. Intermediate variables and processsing, that is not immediately output to the resulting document, may also possible, but as of 24Oct2019 I have not yet implemented that. double "/*", eg in "/*/*$", denotes de-activated commands. This is done automatically for each pass As of 23Oct2019, you must manually re-run "$d_bin""Lucas - formulae & augment.sh" on the created file to do iterative file insertions, for example when processing brings in files that also have commands. (... this should be automated at some time...) This should be automated in the future. +--+ "/*_endCmd" The character sequence "/*_endCmd" at the start of a line simply marks the endpoint of the command initiated by "/*$". It is put in as a convenience to clearly show the inserted portion in the generated txtDoc. Sometimes commands give a simple one line output (such as timestamps, filenames), or perhaps don't write anything into $p_augmented, in which case "/*_endCmd" is omitted. To append the "/*_endCmd" line to many new "/*$" lines, use a kwrite regular expression search-replace, such as : search : /*$(.*)n replace : /*$1n/*_endCmdn Hopefully, if you are not using kwrite, you will have equivalent regular expression searh-replace (if not change text processors!!). Emacs would have it, probably vi as well, although I don't use them. +--+ Table of Contents, list of equations The script "$d_bin""txtDoc insert indexes.sh" inserts the : /*_Insert_Table_of_Contents 153: SUMMARY 236: Lucas's Dedication 264: Introduction 346: I. Basics 350: [Observer, particle, ether] reference frames 379: Galilean transformation of the (observer, particle) reference frames, RFp <=> RFo 410: Generalized ether reference frames 446: Euclidean versus Riemannian geometries 546: Formulations of electrodynamics 551: Maxwell's equations 563: Covariant version 567: ?Heaviside? 4-vector formulation 571: ?Hamilton's? quaternion formulation 577: Lucas's equivalent 581: Ed Dowdye Jr's "Extinction shift principle" 585: Questions 589: Random, scattered questions 598: Initial linearity assumptions, but non-linear models 604: Superluminal speeds 616: Howell's use of the Kahan formulation for a "Scalar derivative of the norm of a vector function" 633: Time delays / Field lag - Temporal equivalence within a frame of reference for "short" distances? 643: How do toroidal [electrons, protons] behave with [spin,rotations, accelerations]? 653: Do toroidal [electrons, protons] [spin, oscillate] via [precession, obliquity]? 661: PROBLEM - When is an induce field "real"? 681: Is my application of Lenz's Law legitimate? 686: Major discrepancies between my own derivations and those of Lucas 690: BTodv(POIo,t)]" 779: BTpdv(POIo(t),t)] without use of Lenz's Induction Law (need to RE-CHECK!!!) 866: Lucas's vt*[cos(Aθpc(POIo(t),t)) - 1] term 875: B X v field is not electrostatic in nature 882: II. Derivations for a POIp = POIo(t) fixed in the particle reference frame (RFp) 887: Basic measures 890: Figure "Basic measures for for the particle reference frame RFp, using POIp=POIo(tx)" 911: Rpcv(POIp), Aθpc(POIp), Aφpc(POIp) are constants 925: Rpcv(POIo(t),t) 979: Rpcs(POIo(t),t) 1069: sin(Aθpc(POIo(t),t)) 1109: cos(Aθpc(POIo(t),t)) 1139: R_O0_pcs(POIo(t),t) 1169: RθPI2pcs(POIo(t),t) 1203: K0, K1, K2 for use in differentiations 1223: K0(t=0), K1(t=0), K2(t=0) in terms of E0ods(POIo,t) for relativistic factor, when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! 1254: EIods(POIo,t=0,ith stage) 1270: K_1st, K_2nd for use in differentiations 1338: K_1st(t=0), K_2nd(t=0), K_3rd(t=0) in terms of E0ods(POIo,t) for relativistic factor, when [t=0, RFp=RFo @t=0], use only AFTER differentiations!!! 1364: E0pdv(POIp) 1380: E0odv(POIo,t) 1392: B0pdv(POIp) = B0odv(POIo) = 0 1404: BIpdv(POIp) = 0 ≠ BIodv(POIp(t),t) = BIodv(POIo,t) 1420: BTpdv(POIp) = 0 1436: EIpdv(POIp) = 0 1444: ETpdv(POIp) = E0pdv(POIp) 1456: Derivatives 1460: Figure "Calculus for RFp, using POIp=POIo(t)" 1465: 1473: 1480: Aθpc(POIp)] = 0 1490: Rpcv(POIo(t) ] 1514: Rpcs(POIo(t),t) ]" 1607: Rpcs(POIo(t),t)] 1780: Rpcs(POIo(t),t)^(-α)] 1800: Aθpc(POIo(t),t)]" 1806: Aθpc(POIo(t),t)] 1911: Rpch(POIo(t),t)]" 1917: Rpch(POIo(t),t) ] 1986: sin(Aθpc(POIo(t),t))] 2098: cos(Aθpc(POIo(t),t))] 2235: Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))] = 0 2363: Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))] 2497: Rpcs(POIo(t),t)^(-b)*sin(Aθpc(POIo(t),t))^a] 2777: Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] 2848: Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] = 0 2923: K0,K2] for use in differentiations 2952: K1] for use in differentiations 2984: K0(t=0),K2(t=0)] in terms of E0ods(POIo,t) for relativistic factor, when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! 3013: K_1st,K_2nd,K_3rd] for use in differentiations 3069: K_1st(t=0),K_2nd(t=0),K_3rd(t=0)] in terms of E0ods(POIo,t) for relativistic factor, when [t=0, RFp=RFo @t=0], use only AFTER differentiations!!! 3101: E0pds(POIp)] 3111: E0ods(POIo,t)] 3137: Summary of ith stage EIods(POIo,t,ith stage)) derivations 3263: Summary of ith stage ETods(POIo,t,ith stage)) 3290: K_1st + f_sphereCapSurf(EIods(POIo,t=0,(i-1) stage))) ] 3302: E0ods(POIo,t)*sin(Aθpc(POIo(t),t))^a] 3359: E0ods(POIo,t=0)*Rpcs(POIo(t),t)^(-β)*sin(Aθoc(POIo))^a] 3502: E0odv(POIo)] 3512: BTodv(POIo,t)] 3528: Integrals 3531: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^z] 3577: Rpcs(POIo(t),t)^(-β)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a)] 3806: III. Derivations for a POIo = POIp(t) fixed in the observer reference frame (RFo) 3811: Basic measures 3814: Figure "Basic measures for the observer reference frame RFo, using POIo=POIp(tx)" 3833: Rocv(POIo), Aθoc(POIo), Aφoc(POIo) are constants in RFo 3851: Rocv(POIp(t),t) 3866: Rocs(POIp(t),t) 3896: RO0ocs(POIp(t),t) 3913: ROPI2ocs(POIp(t),t) = constant 3934: sin(Aθoc(POIp(t),t)) 3955: cos(Aθoc(POIp(t),t)) 4009: R_O0_ocs(POIo) 4018: E0odv(POIo,t) = E0pdv(POIo(t),t) ≠ E0pdv(POIp) = constant, except when t = tx 4083: E0ods(POIo,t) = E0pds(POIo(t),t) ≠ (for t ≠ tx) E0pds(POIp) = constant 4135: Figure "BTodv(POIo,t)" 4142: BIodv(POIo,t) = BIodv(POIp(t),t) ≠ BIpdv(POIp) = 0 4151: BTodv(POIo,t) = BTodv(POIp(t),t) ≠ BTpdv(POIp) = BTpdv(POIo(t),t) = 0 ???? 4164: Prediction of direction of field (B), given that the current I flows in the direction of the thum 4397: EIodv(POIo,t) = EIodv(POIp(t),t) ≠ EIpdv(POIp) = 0 4404: ETodv(POIo,t) = ETodv(POIp(t),t) = E0odv(POIp(t),t) + EIodv(POIp(t),t) ≠ ETpdv(POIp) = 0 4425: ETodv(POIo,t) = ETpdv(POIo(t),t) 4493: Derivatives 4496: Figure "Calculus for RFo, using POIo=POIp(t)" 4515: Aφoc(POIo)] = 0 4532: Rocv(POIp(t),t) ] 4951: 5162: Figure "Electrostatic field basics & calculus for a POIo" 5175: DEFINITIONS 5698: E0pds(POIo(t),t)] - cheating E0ods(POIo,t) scalar approach 5814: BTodv(POIo,t)]" 6229: Lenz's Induction Law - Basics and Calculus 6238: Lenz's Induction Law and it's context 6327: ET_LENZodv(POIo(t),t) = ET_LENZpdv(POIo(t),t), using Lenz's Induction Law 6393: EI_LENZods(POIo(t),t) = EI_LENZpds(POIo(t),t), using Lenz's Induction Law 6461: ET_LENZods(POIo(t),t) = ET_LENZpds(POIo(t),t), using Lenz's Induction Law 6516: BT_LENZodv(POIo(t),t) = BT_LENZpdv(POIo(t),t), using Lenz's Induction Law 6580: EI_LENZpdv(POIo(t),t)] - using Lenz's Law, proper vector approach 6810: ET_LENZpdv(POIo(t),t)], using Lenz's Induction Law 7022: BT_LENZpdv(POIo(t),t)] using Lenz's Induction Law, cheating scalar substitutions 7385: Basics and Calculus using Lucas's results from Thomas Barnes iterations 7389: Thomas Barnes iterations, it's context and relation to Lenz's Induction Law 7415: Howell's correction to the f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) iteration factor 7430: ET_BARNodv(POIo(t),t) = ET_BARNpdv(POIo(t),t), using Lucas's results from Thomas Barnes iterations (needs verification!) 7618: BT_BARNodv(POIo(t),t) = BT_BARNpdv(POIo(t),t), using Lucas's results based on Thomas Barnes iterations (needs verification!) 8442: IV. Expressions for the t=0 reference frame (RFt) 8454: Procedures for consistent reference frame switching 8469: Calculus of RFt 8509: APPENDICES 8512: Future extensions of the Universal Force 8527: Gaussian versus SI units ; 8541: Symbol checking and translation - short description 8627: HFLN = Howells FlatLiner Notation !!!!!!!!!!!!!! 31May2016 8647: Document build short description 8732: REFERENCES /*_Insert_equations at the point where these appear. It is automatically run by "$d_bin""Lucas txtDoc augment.sh". To generate a FULL "Table of Contents" (TOC) [comment, uncomment] the lines below in "$d_bin""txtDoc insert indexes.sh" : # Default - extract the first 3 levels of headings only grep -n "^(>>>){1,3} " "$p_augmented" | sed 's/>>>/ /g' >>"$p_headings_init" # All headings : " #grep -n "^>>>" "$p_augmented" | sed 's/>>>/ /g' >>"$p_headings_init" You can also edit those lines for variants of the TOC. +--+ Viewing files while reading Not all files are inserted in txtDox by the "/*$" command approach. Instead, one might want to view a file during the process of reading, or the file is an [image, spreadsheet] which cannot be directly incorporated in a txtDoc. Therefore, a bash line is provided, which can be copy-pasted into a terminal window to show the file. I have yet to provide default window [size, position] coding, which will be application-specific. Note that a simple "comment" symbol "/*" is used in these cases. - view a text file, in this case using my preferred text editor : /* kwrite "" - more general form, where readers can pre-define an environmental variable in lieu of using a specific program : /*$ eval '$txtEditor ""' - view an image file, again using a specific image viewer : /* eog "" - more general form, where readers can pre-define an environmental variable in lieu of using a specific program : /*$ eval '$imageViewer ""' +--+ More information and practice 24Oct2019 ... for later inclusion here using links rather than inserted text, plus creation of the bash script file ... /*_endCmd /************************************************* >>>>>> REFERENCES /*/*$ cat >>"$p_augmented" "$d_Lucas""context/references.txt" Edward Dowdye 2001 “Discourses & mathematical illustrations pertaining to the Extinction Shift Principle under the electrodynamics of Galilean transformations” copyright 1992, printed by Ed Dowdye, Second edition 2001, ISBN 0-9634471-5-7 Charles William Lucas Jr. 2013 "The Universal Force, Volume 1 : Derived from a more perfect union of the qaxiomatic and empirical scientific methods" (c) Charles W. Lucas, www.commonsensescience.org ISBN-13: 978-1482328943 Bill Lucas NPA presentation slides : Electrodynamic Origin Force of Gravity (F = Gm1m2/r2) www.worldsci.org/pdf/slides/abstract_slides_6539.pptx Empirical Equations of Electrodynamics for v=constant. Ampereps Law. Faradays Law . Gauss Electrostatic Law. Gauss Magnetostatic Law . Lenz's Law John David Jackson 1999 "Classical Electrodynamics, 3rd Edition", John Wiley & Sons, 808pp, ISBN 978-0-471-30932-1 Erwin Kreyszig 1972 "Advanced Engineering Mathematics, Third Edition" John Wiley & sons, ISBN 0-471-50728-8 COOL! https://en.wikipedia.org/wiki/Oliver_Heaviside "... Oliver Heaviside FRS[1] (/ˈɒlɪvər ˈhɛvisaɪd/; 18 May 1850 – 3 February 1925) was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques for the solution of differential equations (later found to be equivalent to Laplace transforms), reformulated Maxwells field equations in terms of electric and magnetic forces and energy flux, and independently co-formulated vector analysis. Although at odds with the scientific establishment for most of his life, Heaviside changed the face of telecommunications, mathematics, and science for years to come.[2] ..." /*_endCmd enddoc