www.BillHowell.ca's review of Bill Lucas's math in his book the "Universal Force, Volume I"
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SUMMARY
This file contains my own stepby step derivations of equations in Chapter 4 of Charles W. Lucas's book "The Universal Force, Volume 1". Discrepancies between Lucas's results and my own are far more likely due to my own errors than Lucas's, but there is a chance that several minor errors in Lucas's book will have been highlighted.
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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, stepbystep documentation, that is easily [auditable, editable, extendable], abeit in a specialised nonstandard (yet more precise than conventional) system of [variable notations, symbols, notations, format].
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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 autoindents to the last tabbed position of a line
 when necessary, use full screen mode to more easily see very long equations.
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TABLE OF CONTENTS
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202: Lucas's Dedication
230: Introduction
312: Lucas 4  Derivation of the Universal Electrodynamic Force Law for constant velocity
316: 4.1  Proper Axioms of Electrodynamics
342: Fundamental Equations Of Electrodynamics ;
623: Howell  Appendix A, Derivation of the BiotSavart and Grassman form of Amperes Law
637: 4.2  Derivation of Electrodynamic Force Law
2304: Summary  Derivation of the relativistic correction factor (1  β^2*sin(Aθpc(POIo(t),t=0))^2)^(3/2)
2351: Highly restrictive conditions for dropping the (cos  1) terms /%
2372: Targeted results /%
2391: Multiple conflicting hypothesis /%
2404: Does the relativistic correction factor matter? /%
2506:
2778: Bottomup (2b1)
4192: Lucas 5  Extension of the Universal Force Law to include acceleration
4198: 5.1  Generalized electromagnetic potential U(r,v)
4445: 5.2  Acceleration fields and radiation ;
4647: Lucas 6  Extension of the Universal Force Law to include radiation reaction da/dt
4651: 6.3  Derivation of nonrelativistic radiation reaction force ;
4768: 6.4  Derivation of relativistic radiation reaction force ;
4832: Lucas 7  Electrodynamic origin of gravitational forces
4836: 7.1  Introduction, Electrodynamic origin of gravitational forces ;
4980: 7.2  Origin of gravitational forces ;
5106: 7.3  Computation of radial force term ;
5431: 7.4  Corroborating evidence for radiative decay of gravity ;
5509: 7.6  Computation of nonradial gravitational force term ;
5677: 7.8  Origin of Hubbles Law due to gravitational redshifts ;
5700: Lucas 8  Electrodynamic origin of Inertial forces
5703: 8.1  Introduction ;
5808: 8.2  Derivation of force of inertia from Universal Force Law ;
5921: 8.3  Derivation of Newtons 2nd law from 1st acceleration term ;
6180: 8.4  Additions to Newtons 2nd law from 2nd acceleration term ;
6449: Lucas 9  Structure and harmony of the universe
6452: 9.1  Structure is from symmetry of the Universal Force ;
6506: Lucas 10  Machs principle and the concept of mass
6509: 10.1  Inertial mass ;
6564: 10.2  Gravitational mass ;
6738: APPENDICES
6741: Future extensions of the Universal Force
6756: Gaussian versus SI units ;
6770: Symbol checking and translation  short description
6856: HFLN = Howell's FlatLiner Notation !!!!!!!!!!!!!! 31May2016
6876: Document build short description
6961: REFERENCES
EQUATIONS :
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352: (401) Generalized_Amperes_Law
403: (402) Faradays_Law
445: (403) Gauss_Electrostatic_Law
479: (404) Gauss_Magnetostatic_Law
508: (405) Lenz_Induction_Law
528: (405a) E as sum of E0 & Ei
545: (406) Lorentz_Force_Law
595: (407) Galilean_transformation
645: (408) Induced_magnetic_flux_density from Amperes law
683: (409) Frame_transformation_info_lost by Maxwell
726: (410) Galilean_transformation
740: (411) E&B_fields_static_plus_induced
762: (412) E Galilean transformation particle to observer frames
777: (413) Total B magnetic flux density as induced from E0 + Ei
842: (414) B&E point charge  substituted Amperes law
918: (414a) Point particle and symmetry
942: (415) E,B for symmetry point charge @v_const
1040: (416) E,B for symmetry point charge @v_const
1138: (417) Spherical coordinate transforms
1177: (418) Changing magnetic flux linked by a circuit proportional to induced E field around the circuit
1211: (419) E,B for symmetry point charge @v_const  Stokes theorem
1297: (420) Convective_derivative
1321: (421) convective derivative of Total magnetic flux density Bi
1443: (422) KelvinStokes integration of convective derivative of Bi total
1474: (423) Faradays_Law_for_rest_circuit integral form E,B
1549: (424) E&B for [Faradays + part/obs frameTrans]  towards FU_Faradays_Law
1631: (425) E&B for [Faradays + part/obs frameTrans]  towards FU_Faradays_Law
1665: q,E,Bi)
1709: (427) Lorentz Force
1891: (428a) Faradays_law_spherical_coords  ∇´Ei(ro  vo*t,t) term
1968: Bi(ro  vo*t,t)] term
2003: (429a) Faradays_law_spherical_coords  1st term
2088: (429b) Faradays_law_spherical_coords  2nd term
2110: (430) Faradays_law integrated over θ
2256: (431) From Lenzs law and symmetry of local forces
2286: (431a) Machs principle  Lenz works, SRT & covariant Maxwell fail
2414: (432) EIods(POIo,t=0,1st stage), F therefore E balance  iteration #1 on (430)
2591: (433) EIods(POIo,t=0, 2nd stage), F therefore E balance
2719: (434) EIods(POIo,t=0, 2nd stage), K_2nd from taking partial derivatives wrt time
2889: (435) E0ods(POIo,t) truncated expression with ONLY E0ods(POIo,t) terms
3050: (436) ETods(POIo,t) expression with ONLY ETods(POIo,t) terms
3085: (437) Er and the binomial series, leading to the relativistic correction factor
3668: (438) Binomial_expansion_for_E0_terms
3711: (439) E(r,v) for constant velocity, nonpoint charge, observer reference frame
3775: (440) Gauss_Electrostatic_Law
3793: (441) L(v) expression for Gauss law for electric charge
3893: (442) Special integral with binomial series (1  b^2*sin^2(O))^(3/2)
3972: (4_43) E&B_fields_self_consistent
4022: (444) F_total by moving charge distribution on a test charge q'
4125: (445) Vector identities for Lorentz Force derivation
4150: (446) Vector_operations used for the Lorentz force
4202: 05_01 ;
4235: 05_03 ;
4380: 05_08 ;
4449: 05_11 ;
4468: 05_12 ;
4487: 05_13 ;
4840: 07_01 ;
4861: 07_02 ;
4903: 07_04 ;
5015: 07_08 ;
5057: 07_09 ;
5133: 07_12 ;
5192: 07_14 ;
5475: 07_24 ;
5513: 07_25 ;
5681: 07_29 ;
5753: 08_03 ;
5772: 08_04 ;
5792: 08_05 ;
6621: 10_05 ;
6638: 10_06 ;
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.
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waiver, copyright
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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.
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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 nonthirdparty content of this document under either:
The GNU Free Documentation License (http://www.gnu.org/licenses/); with no Invariant Sections, FrontCover Texts, or BackCover Texts.
Creative Commons AttributionNoncommercialShare 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.
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>>> Lucas's Dedication
(This is copied directly from his book.)
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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 AndreMarie Ampere in emphasizing the role of fields in extending ther electRodynamic force to great distances to replace Weber's actionatadistance 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 finitesized 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.
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>>> Introduction
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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 reformulation 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 LorentzPoicare'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 bandaids 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 wellresearched 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 stepbystep 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 nonstandard 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"
Peerreview 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  MetaLevel 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 nonbeliever 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 longforgotten [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 :
(41) Generalised Amperes Law
(45) Lenzs Induction Law
(46) 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 stepbystep retyping 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 stepbystep. 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 uptodate 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
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>>> Lucas 4  Derivation of the Universal Electrodynamic Force Law for constant velocity
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>>>>>> 4.1  Proper Axioms of Electrodynamics
p63h0.2 "... In the derivation that follows [2,3,4] the approach is taken that a proper electrodynamic force law should be compatible with the following set of axioms which is more complete than that of Maxwell and corresponds better with reality :
Lucas_ED_axioms :=
1. Coulombs law for the force between static charges
2. Amperes generalized law for the force between current elements
3. Faradays law of electromagnetic induction
4. Gausss laws
5. Only contact forces exist in nature
6. Lenzs law for induction
7. Finitesize of charged particles with interior structure
8. Fields of charges remain attached when charges move and have tensile strength
9. Galilean invariance
10. Newtons third law  for every action there is an equal and opposite reaction
11. Conservation of kinetic and radiant energy
12. Conservation of momentum (radiation reaction etc)
13. Machs principle that local physical laws are determined by the largescale structure of the universe
14. Nonlinear electrodynamic processes occur in lasers and other phenomena
Note that axioms 514 are missing in the relativistic version of electrodynamics based on Maxwells equations. ..."
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>>>>>>>>> Fundamental Equations Of Electrodynamics ;
p63h1.0 "... The fundamental equations of electrodynamics are based upon six empirical laws that are valid for constant velocity, i.e. Note that both Amperes law and Faradays law involve the observers reference frame and the primed moving frame of reference that are described by the Galilean transformation of Lucas04_07.
17Aug2015 Howell  variables t, c, n q are scalar, the rest are vector. Also, the primes (´) indicate the moving frame of reference, and unprimed is the observers frame.
21May2016 Howell  There are big differences between Lucas's six empirical laws, and classical conventional expressions (eg Maxwells equations). I have noted that in my comments for equations (41) through (46) below, and I will revisit Lucas's justifications in Appendix A etc later. For this firstpassthrough,as with other problematic situations, I simply follow on with Lucas's [framework, basis] as a check on the [consistence, correctness] of derivations that flow from it.
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/*> (401) Generalized_Amperes_Law
Equation (41) is DIFFERENT from Maxwell equation equivalent!
 Doesn't have the curl operation on B, rate of change of E0.
Also  Lucas's dividing by c, maybe μ0,ε0 are problems too, 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
reference Lucas p65h0.5 This is shown in (58), derived in Appendix A
reference : https://en.wikipedia.org/wiki/Amp%C3%A8re%27s_circuital_law
MaxwellAmpere circuital equation, integral form, C is a closed curve :
/$ ∮[∂(l): B) = ∬[•dArea: μ0*J + μ0*ε0*∂[∂(t): ET]]
/% ∮[∂(l): B) = ∬[•∂(Area): μ0*J + μ0*ε0*∂[∂(t): E]]
/* differential form :
/$ ∇B = μ0*J + μ0*ε0*∂(ET)
/% ∇B = μ0*J + μ0*ε0*∂(E)
/*"In cgs units" ????
integral form :
/$ ∮[∂(l): B) = 1/c*∬[•dArea: 4*π*J + ∂[∂(t): ET]]
/% ∮[∂(l): B) = 1/c*∬[•∂(Area): 4*π*J + ∂[∂(t): E]]
/*differential form :
/$ ∇B = 1/c*[4*π*J + ∂[∂(t): ET]]
/% ∇B = 1/c*[4*π*J + ∂[∂(t): E]]
/* NOTE : It seems to be implied in the wikipedia article that
[B, E] are functions of (r,t), as explicitly shown in Lucas's expressions!
Lucas (47) below states t = t
Lucas's use of the "Generalized Amperes Law" is NOT the same as the Maxwell equation equivalent.
This is a major point that he is making.
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn 04_01  Generalized_Amperes_Law
/$ B(r´,t´) = v/c × E0(Rpcv,t´)
∇B = 1/c*[4*π*J + ∂[∂(t): ET]]
(mathL) β = Vons(PART)/c
(endMath)
/% BIodv(POIo,t) = v/c × E0pcv(POIp,t))
∇BTodv(POIo,t) = 1/c*[4*π*Jodv(POIo,t) + ∂[∂(t): E0pdv(POIp)]]
/* NOTE  I have to check Appendix A later...
DIFFERENT from Maxwell equation equivalent! : Doesnt have the curl operation on B, rate of change of E0.
/**********************************************************
/*> (402) Faradays_Law
ref : https://en.wikipedia.org/wiki/Faraday%27s_law_of_induction
Maxwell–Faraday equation in integral form by KelvinStokes theorem :
integral form, C is a closed curve :
/$ ∮[•∂(l),.over.∂Σ: ET) = ∬[•dArea ,.over.: ∂[∂(t): B]]
/% ∮[•∂(l),.over.∂Σ: E ) = ∬[•∂(Area),.over.: ∂[∂(t): B]]
/*differential form :
/$ ∇ET(r,t) = ∂[∂(t): B(r,t)]
/% ∇E (r,t) = ∂[∂(t): B(r,t)]
/*where : ∂Σ is a surface bounded by the closed contour ∂Σ,
Recast integral form using
/$ B = ∬[dArea: B(r,t)•n], B(r,t) for B (same for ET)
/% B = ∬[∂(Area): B(r,t)•n], B(r,t) for B (same for E)
/* Still need to check the primes, etc...
 looks similar except for 1/c and lack of a curl operator for E.
however, "cgs units" seem to haver pulled a trick in another example???
BUT  important point is the reversal of the integration/derivative operators!!!
THIS SEEMS TO BE A HUGE ERROR  local distributions may average to zero,
but could have extremely important local effects even if overall they net out?
Also  should have dot product of E & dl !! (implied by Lucas for vector multiplication unless specified otherwise?)
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn Problem_or_challenge
04_02
Faradays_Law
/$ ∫[•∂(l): ET(r´,t´)] = 1/c*∂[∂(t): ∫[∂(Area): B(r,t) •n]]
∫[•∂(l): ET(r´,t´)] =  ∫[∂(Area): ∂[∂(t): B(r,t))]•n]
/% ∫[•∂(l): ETodv(POIp(t),t)] = 1/c*∂[∂(t): ∫[∂(Area): BTodv(POIo,t) •n]]
∫[•∂(l): ETodv(POIp(t),t)] =  ∫[∂(Area): ∂[∂(t): BTodv(POIo,t))]•n]
/* (OK  but DIFFERENT from Maxwell equation equivalent! : missing curl(E). Does this give good results for experimental data?
another important difference is the reversal of the integration/derivative operators!!
/**********************************************************
/*> (403) Gauss_Electrostatic_Law
nh is a unit vector normal to the surface dA
https://en.wikipedia.org/wiki/Electrostatics
integral form :
/$ ∬[•dArea : ET] = 1/ε0*Qenclosed = ∫[•∂^3r,.over.V: ρ/ε0)
/% ∬[•∂(Area): E ] = 1/ε0*Qenclosed = ∫[•d^3r,.over.V: ρ/ε0)
/* where d^3r = dx*dy*dz
differential form via the divergence theorem :
/$ ∇•ET = ρ/ε0
/% ∇•E = ρ/ε0
/* Use the integral form, with
/$ •∂(Area) replaced by •nh*∂(Area)
ET(r,t) for ET
4*π*Q(particle) for ∫[•∂^3r,.over.V: ρ/ε0)
/*> again, which variables have ε0, μ0, π or not, Ill have to check later
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
04_03
Gauss_Electrostatic_Law
/$ ∫[∂(Area): ET(r,t)•nh] = 4*π*Q(particle)
/% ∫[∂(Area): E0pdv(POIp)•Roch(POIo)] = 4*π*Q(particle)
/* OK  same as conventional expression
again, which variables have ε0, μ0, π or not, Ill have to check later
/**********************************************************
/*> (404) Gauss_Magnetostatic_Law
/*# https://en.wikipedia.org/wiki/Magnetostatics
Gauss's law for magnetism (steady state!)
integral form :
/$ ∮[•∂(Area),.over.S: B) = 0
/% ∮[•∂(Area),.over.S: B) = 0
/*differential form :
/$ ▽•B = 0 ;
/% ▽•B = 0 ;
/*reference : Jackson 1999 p179h0.2
/$ ▽•B = 0 ;
/% ▽•BTpdv(POIp,t) = ▽•BTodv(POIo,t) = 0 (BIpdv(POIp) = 0 always)
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
04_04
Gauss_Magnetostatic_Law
/$ ▽•B = 0 ;
/% ▽•B = 0 ;
/* (OK  piece of cake...
remember  this is for no charge accumulation, steady state electric, magnetic fields!!
/**********************************************************
/*> (405) Lenz_Induction_Law
/* reference : https://en.wikipedia.org/wiki/Lenz%27s_law
Wikipedia mentions Lenzs law as being qualitative,
relating to direction of Faradays law. But they also say :
"... Lenzs law /ˈlɛnts/ is a common way of understanding
how electromagnetic circuits obey Newtons third law and the conservation
of energy.[1] ..."
This concurs with oftrepeated statements by Lucas
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Question
04_05
Lenz_Induction_Law
/% EIodv(POIp(t),t) ∝ E0odv(POIo,t) = lambda(Vonv)*E0odv(POIo,t)
/* (Qualitatively OK  but I can't seem to find support for this specific form.
/**********************************************************
/*> (405a) E as sum of E0 & Ei
This is very important, especially as Lucas suggests that the static
and induced fields behave differently (eg induced fields interact NONLINEARLY?
Ill have to check later. (Hooper & other references from Lucas)
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
04_05a E as sum of E0 & Ei
/$ ET(r,v,t) ∝ E0(r,t) = λ(v)*E0(r,t)
/% ETodv(POIo,t) = E0odv(POIo,t) + EIodv(POIo,t)
/* Didnt look for this yet
I havent looked for this yet, but Jackson shows "Normal" linear super´osition, except solid materials, extreme conditions.
/**********************************************************
/*> (406) Lorentz_Force_Law
NOTE  This looks WRONG compared to (406), but I think it is (406) that is WRONG!
25Aug2015 Lucas04_6 Lorentz_force_law : missing a q in the second term?
see p64h0.5 Lucas04_6 versus p69h0.65 Lucas (426)  in the latter the
charge appears in both terms :
Lucas04_26 := Lorentz_force_
/$ F(r  v*t,t) = q*ET (r  v*t,t) + q/c*[vBi(r  v*t,t)] ;
/%^% F(ro  vo*t,t) = Q(particle)*E (ro  vo*t,t) + Q(particle)/c*[Vonv(PART)BIodv(ro  vo*t,t)] ;
/* Note that in Lucas (426) Lucas DERIVES the Lorentz force simply from the Universal force,
so he suggests it is NOT a fundamental force.
also WRONG?  dividing by c, negative sign rather than positive
Note : The "c" in Gaussian units is a persistent source of my errors, but Ill leave it be.
reference : Jackson 1999 p557h0.7 Equation (11.144) covariant form of
Lorentz force.. This section discusses tensors of 4th order
/$ ∂/dτ(p^α) = m*∂/dτ(U^α) = q/c*F^(α*β)*U_b
/* wow  not straightforward identifying Lorentz force law in Jackson's book!!
reference : https://simple.wikipedia.org/wiki/Lorentz_force
/$ F = q*ET + q*vB
/%^% F = Q(particle)*E + Q(particle)*Vonv(PART)B
/* This agrees with (426), but disagrees with (406)
same issue with : https://in.answers.yahoo.com/question/index;_ylt=AwrTcdYh5.hV8uQAuVUXFwx.;_ylu=X3oDMTBzcnZmYjNuBHNlYwNzcgRwb3MDMTAEY29sbwNncTEEdnRpZAM?qid=20091114100116AA701h6
reference : http://www.mae.ncsu.edu/buckner/courses/mae535/VanDenBroeck.pdf
also has q with B
Below I'll add in the (r,v,t) etc
add_eqn "Lucas_typo_or_omission
04_06
Lorentz_Force_Law
/$ F(r,v,t) = q* ET(r,v,t)  v/cB(r,v,t)
/%^% FLENZodv(POIo,t) = Q(particle)*[ ETodv(POIo,t) + Vonv(PART)BTodv(POIo,t) ]
/* (WRONG (typo probably) Lucas is missing the "q" in q*vB(r,v,t) see (406) versus (426) and web references
+
Dimensionality check (see also "Howell  Variables, notations, styles for Bill Lucas, Universal Force.odt"
... For now, I havent got this to work!!!!!!!!
/**********************************************************
/*> (407) Galilean_transformation
This is straightforward, so I wont check.
(eg Jackson 1999 p515h0.55 Eq (11.1)
07Jan2016 Actually  this caused a lot of trouble and wasted work, as earlier
assumptions about geometry turned out to be incorrect.
A key issue is that the transformation DEPENDS ON the [particle, observer]
reference frames to be EXACTLY ewqual at time t=0, after which the particle
frame moves with vo*t.
My attempts to generalize the geometry (rotation, translation of observer frame,
led to hideousy complex expressions. Of course, Jacobians would handle that,
but by hiding detail to make it "look simple".
In essence, the simple, powerful expressions of physics are only such for
braindead system setsups.
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
04_07
Galilean_transformation
/$ Rpcv = r  v*t and t´ = t
/%^% Rpcv = ro  vo*t and t´ = t
/* OK  straightforward (eg Jackson 1999 p515h0.55 Eq (11.1)
/*************************
>>>>>>>>> Howell  Appendix A, Derivation of the BiotSavart and Grassman form of Amperes Law
This is in a separate file!!
p178 Appendix A : Derivation of the BiotSavart and Grassman
Form of Amperes Law
/$LA_01 Fij = i*i´* [ r*(dsi•dsj´)*φ(r) + dsi*(dsj´•r)*P2(r) + dsjp*(dsi•r)*P3(r) + r*(dsi•r)*(dsjp•r)*ψ(r) ] ;
/% Fij = i*i´* [ r*(dsi•dsj´)*P(r) + dsi*(dsj´•r)*P2(r) + dsjp*(dsi•r)*P3(r) + r*(dsi•r)*(dsjp•r)*ψ(r) ] ;
/* This has to wait for some time before I can get to it.
I will work on the front part of the book first.
/********************************************** ;
>>>>>> 4.2  Derivation of Electrodynamic Force Law
p64h0.9 "... ..." :
/**********************************************************
/*> (408) Induced_magnetic_flux_density from Amperes law
Lucas p64
/$401 Bi(r,v,t) = (v/c)E0(r,t)
/%^% BIodv(POIo,t) = (Vonv(PART)/c)E0odv(POIo,t)
/* The Grassman form of the generalized Ampere force law is based on derivations in Appendix A (eq (A19).
(408) is the derivation of (401) from the Grassman/BiotSavart form of Amperes Law
This is derived in Appendix A...
/$ q/c*(vr´)/rs^3 = (v/c)E0(r,t)
/%^% Q(particle)/c*(Vonv(PART)Rpcv(POIp))/Rocs(POIo)^3 = (Vonv(PART)/c)E0odv(POIo,t)
/* reference : Jackson 1999 p176h0.15 Equation (5.5):
/$ B = k*q*(vx )/x ^3
/%^% B = k*Q(particle)*(Vonv(PART)x )/x ^3
/* replace x with r, and note that this agrees with the "q/c" part of (48) :
/$ B = q*k*(vr´)/r´s^3 ;
/%^% B = Q(particle)*k*(Vonv(PART)Rpcv(POIp))/Rpcs(POIp)^3 ;
/* But should this be Ei, not E0? That would give you :
/$ Bi(r,v,t) = q/c*(vr´)/rs^3 = (v/c)E0(r,t)
/%^% BIodv(POIo,t) = Q(particle)/c*(Vonv(PART)Rpcv(POIp))/Rocs(POIo)^3 = (Vonv(PART)/c)E0odv(POIo,t)
/* Additional issue :
p66h0.2 "... Thus it was logically inconsistent to introduce retardation
effects into classical electrodynamics for nonrediation situations. ..."
For Lucas, maybe, but it IS a legitimate concern!?!
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
04_08
Induced_magnetic_flux_density from Amperes law
/$ Bi(r,v,t) = (v/c)E0(r,t)
/% BIodv(POIo,t) = (Vons(PART)/c)E0pdv(POIp)
/* OK  Must check derivation of Appendix A Equation (A19)
/**********************************************************
/*> (409) Frame_transformation_info_lost by Maxwell
Lucas  p65h0.85 "... Note that if Amperes law, as represented by equation (48),
is cast into its usual Maxwell form of equation (49), the reference frame transformation
information is lost. ..."
reference : Jackson 1999
p179h0l.75 Equation (5.21)
/$ ∇B = μ0*J(x´)  μ0*/4/π*∇∫[∇´•J(x´)/x  x´]*∂^3(x)
/% ∇B = μ0*J(x´)  μ0*/4/π*∇∫[∇´•J(x´)/x  x´]*d^3(x)
/*This is very similar to Lucas expression, but there are a couple of problems :
1. units  I can swap μ0 & c, but will end up with ε0 in there somewhere
This looks like an error by Lucas, but also may be due (partially or fully)
to unfamiliar unit issues (See "Gaussian vs SI")
2. 3rd derivative of x vs 1st derivative of r (???)
Reexpress Jackson's (5.20) with r instead of x :
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Lucas_typo_or_omission
04_09
Frame_transformation_info_lost by Maxwell
/$ ∇B(r,v,t) = 4*π/c*J(r,t) + 1/c *∇∫[∂(r): ∇•J(r,t)/r  r´}
≈ 4*π/c*J(r,t)
/%^% ∇BTodv(POIo,t) = 4*π/c*J(Rocv(POIo),t) + 1/c *∇∫[∂(Rocv(POIo)): ∇•J(Rocv(POIo),t)/Rocv(POIo)  Rpcv(POIp)}
≈ 4*π/c*J(Rocv(POIo),t)
/* when second term is ignored
/$ ∇B = μ0*J(r´) + μ0/4/π*∇∫[∂^3r: ∇•J(r,t)/r  r´}
∇B = μ0*J
/% ∇BTodv(POIo,t) = 4*π/c*Jodv(POIo,t) + 1/c *∇∫[∇•Jodv(POIp,t)/Rpcs(POIo(t),t)  Rpcs(POIp)]∂(r)
∇BTodv(POIo,t) = μ0*Jpdv(POIp,t)
/* for steadystate magnetic phenomena!
(NOT the same!! is this d^3 a typo in Lucas? Lucas versus Jackson  third derivative d^3r´, my typical problem with [c,μ0,ε0,4,π]
NOTE : The second expression is for steadystate magnetic phenomena!
r  r´ term is intriguing => Vons(PART)*t? NOO!!!  special formulation needed
/**********************************************************
/*> (410) Galilean_transformation
Lucas04_10 := Lucas04_07 ;
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
04_10
Galilean_transformation
see Lucas04_07
see Lucas04_07
OK  repeat statement, no need to recheck
/**********************************************************
/*> (411) E&B_fields_static_plus_induced
04Sep2015  As I currently understand it, only the E0 versus Ei
distinction is critical as the later is nonlinear superposition?
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
04_11
E&B_fields_static_plus_induced
/$ ET(r,v,t) = E0(r,t) + Ei(r,v,t)
B(r,v,t) = B0(r,t) + Bi(r,v,t)
/% ETodv(POIo,t) = E0odv(POIo,t) + EIodv(POIo,t)
BTodv(POIo,t) = B0odv(POIo,t) + BIodv(POIo,t)
/* critical distinction between static & induced  have to check later
OK  but where is the effect of the critical distinction between static & induced  have to check later
where B0odv(POIo,t) = 0 in Chapter 4
/**********************************************************
/*> (412) E Galilean transformation particle to observer frames
follows from Lucas's notation see (407)
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
04_12
E Galilean transformation particle to observer frames
/$ ET(r´,t´) = ET(r  v*t,t)
/% (ETpdv(POIp)=E0pdv(POIp)) = ETodv(POIo,t)
/* (OK, EASY  key point, I need to research results and opinions, seems correct
/**********************************************************
/*> (413) Total B magnetic flux density as induced from E0 + Ei
/*/*$ cat >>"$p_augmented" "$d_augment""04_13 work.txt"
Here I will use Howells form of (48) :
25Aug2015 restart equation entry at Lucas04_13
/$ 04_08 Bi(r,v,t) = (v/c)E0(r,t) ;
Check 04_08 Bi(r,v,t) = (v/c)Ei(r,t) ;
/%^% BIodv(POIo,t) = (Vonv(PART)/c)E0odv(POIo,t) ;
BIodv(POIo,t) = (Vonv(PART)/c)EIodv(Rocv(POIo),t) ;
/* This still leaves the static component of B as per (411)
/$ B(r,v,t) = B0(r,t) + Bi(r,v,t)
/%^% BTodv(POIo,t) = B0odv(POIo,t) + BIodv(POIo,t)
/****where B0 is dropped as it "... is not electrostatic in nature ..." (p67h0.4)
02Jan2016 Note : There is no magnetic particle in the system, so B0 = 0
Actually Im confused p67h0.3
PROBLEM  When is an induce field "real"? see the 6 equations in Lucas p64
Generalized Amperes 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 highspeed 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 didnt think of these things. Reminds me that a constant current
in a wire has a B field, but no E field (which I hadnt thought of)!
Use same functional form for B0 & Bi? This has to be verified and justified!
given the induced versus static B fields.
/$ B(r´,t´)
= (v/c)E0(r´,t´) + (v/c)Ei(r´,t´)
= (v/c)[ E0(r´,t´) + Ei(r´,t´) ]
/%^% BTodv(POIp(t),t)
= (Vonv(PART)/c)E0odv(Rpcv(POIp),t´) + (Vonv(PART)/c)EIodv(Rpcv(POIp(t),t)
= (Vonv(PART)/c)[ E0odv(Rpcv(POIp),t´) + EIodv(Rpcv(POIp(t),t) ]
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Question
04_13rev1
Total B magnetic flux density as induced from E0 + Ei
/$ B(r´,t´) = Bi(r  v*t,t) = (v/c)[ E0(r´,t´) + Ei(r´,t´) ]
/%^% BTdv(POIp(t),t) = Vonv(PART)/c[ E0pdv(POIp) + (EIpdv(POIp)=0) ] ???
/* ( OK simple, but I have questions. 10Jan2016
I assume B0 doesn't appear as there is no "other background" B source 
eg permanent magnet of other [charges, currents]. Or is B0 from E0?
PROBLEM  When is an induce field "real"?  depends on relative velocities
so different observers at different relative velocities
see different B fields at the same Point of Interest (POI)!!??
/*_endCmd
/**********************************************************
/*> (414) B&E point charge  substituted Amperes law
/*/*$ cat >>"$p_augmented" "$d_augment""04_14 work.txt"
First term
This expression seems to violate the statements in the previous paragraph!?
"... linear superposition does not hold for the Bv generated fields ..."
Magnetic field arising from single moving point charge :
https://en.wikipedia.org/wiki/Biot%E2%80%93Savart_law
from Maxwells equations :
/$ ET = q/4/π/ε0*(1  v^2/c^2)/(1  v^2*sin(θ´)^2/c^2)^(3/2)*r´h/r´^2
B = 1/c^2*vET
/%^% E = Q(particle)/4/π/ε0*(1  Vonv(PART)^2/c^2)/(1  Vonv(PART)^2*sin^2(Op)/c^2)^(3/2)*rhp/rp^2
B = 1/c^2*Vonv(PART)E
/* where rhp is the unit vector pointing from the current (nonretarded)
position of the particle to the point at which the field is being measured,
and Op is the angle between v and rp.
for v^2 << c^2
/$ ET = q/4/π/ε0*r´h/r´^2
B = μ0/4/π*q*vr´h/r´^2
/%^% E = Q(particle)/4/π/ε0*rhp/rp^2
B = μ0/4/π*Q(particle)*Vonv(PART)rhp/rp^2
/* Jackson 1999 p782h0.15 Table 3 provides conversions
(here to Lucas's Gaussian units, except p180h0.45 under (A11) he uses CGS units?))
/$ ET = 1/4/π/ε0*q *r´h/r´^2 * (4*π*/ε0)^(0.5) = (4*π*ε0)^0.5*q *r´h/r´^2 ??? */???
B = μ0/4/π *q*vr´h/r´^2 / (4*π/μ0)^(0.5) = (μ0/4/π)^ 0.5*q*vr´h/r´^2
/%^% E = 1/4/π/ε0*Q(particle) *rhp/rp^2 * (4*π*/ε0)^(0.5) = (4*π*ε0)^0.5*Q(particle) *rhp/rp^2
B = μ0/4/π *Q(particle)*Vonv(PART)rhp/rp^2 / (4*π/μ0)^(0.5) = (μ0/4/π)^ 0.5*Q(particle)*Vonv(PART)rhp/rp^2
/* NUTS  "units" dont work out (μ0,4,π,ε0)
Here, B = B0(r´,v´,t´), and substituting c for (μ0/4/π)^0.5 as "patch"
until I understand the conversions better
/$ B0(r´,t´) = q/c*(vr´h)/r´^2
/%^% B0(Rpcv(POIp),t´) = Q(particle)/c*(Vonv(PART)rhp)/rp^2
/* Second term vovEi(r´,t´)
immediately, the second term is wrong, as "/c" has been dropped,
or (413) is wrong.
Notice that this second term is a consequence of the [static, induced]
split of [E,B], as per (411) which I still have to check. Use :
/$ v/cEi(r´,t´)
/%^% Vonv(PART)/cEIodv(Rpcv(POIp(t),t)
/* Combined expression
/$ B(r´,t´) = q/c*(vr´)/r´´^2 + v/cEi(r´,t´)
/%^% BTodv(POIp(t),t) = Q(particle)/c*(Vonv(PART)Rpcv(POIp))/r´´^2 + Vonv(PART)/cEIodv(Rpcv(POIp(t),t)
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Lucas_typo_or_omission
04_14
B&E point charge  substituted Amperes law
/$ B(r´,t) = q/c*(vr´h)/r´s^2 + v Ei(r´,t´)
B(r´,t) = q/c*(vr´h)/r´s^2 + v/cEi(r,t)
/%^% B(Rpcv(POIp),t) = Q(particle)/c*(Vonv(PART)rph)/rps^2 + Vonv(PART) EIodv(Rpcv(POIp(t),t)
BTpdv(POIp,t) = Q(particle)/c*(Vonv(PART)Rpch(POIo(t),t))/Rpcs(POIo(t),t)^2 + Vonv(PART)/c(EIpdv(POIp)=0)
/* OK  simple, but Lucas is missing c in 2nd term (check Jackson) 10Jan2016
Lucas's units don't balance! (uses Gaussian units...)
/*_endCmd
/**********************************************************
/*> (414a) Point particle and symmetry
This is stated based on symmetry of field around a single moving charge,
to which the inuced field is attached.
Also rh´=r´/r´, ie. unit vector in direction of r
All this does is convert a scalar (magnitude of Ei) to a vector, but
why was a scalar to begin with?
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
04_14a
Point particle and symmetry
/$ Ei(r´,t´) = Ei(r´,t´)*r´/r´
Ei(r´,t) = Ei(r´,t´)*r´/r´
/%^% EIodv(Rpcv(POIp(t),t) = EIodv(Rpcv(POIp(t),t)*Rpcv(POIp)/Rpcv(POIp)
(EIpdv(POIp)=0) = (EIpds(POIp)=0)*Rpch(POIo(t),t)
/* OK, simple
/**********************************************************
/*> (415) E,B for symmetry point charge @v_const
Adapted non[derivative, integral] form of Amperes law
/*/*$ cat >>"$p_augmented" "$d_augment""04_15 work.txt"
proceeding from Howells (414) because Lucas (414) is missing 1/c term,
and its B(rov,t), NOT B(r´,t)
/$414) B(r´,t´) = q/c*(vr´h)/r´s^2 + v/cEi(r´,t´)
/%^% B(rov,t) = Q(particle)/c*(Vonv(PART)rph)/rps^2 + Vonv(PART)/cEIodv(Rpcv(POIp(t),t)
/*selectively substitute for (rov,t), r´, vov X rph, vov X Ei(ro  vo*t,t) :
as rph & Ei are collinear direction is normal to v,Ei > (vr)h = Pph :
Equation (417) below explains the transformations
/$ r´s = r´ = r  v*t
= [ (Rocs(POIo)*sin(θ))^2 + (Rocs(POIo)*cos(θ)  vs*t)^2 ]^(1/2)
v X r´h = vs*sin(θ´)*φ´hat
v X r´h = vs*sin(θ´)*φ´hat
v X Ei(r  v*t,t) = vs*sin(θ´)*Ei(r  v*t,t)*φ´hat
r´s*sin(θ´) = Rocs(POIo)*sin(θ)
/%^% rps = Rpcv(POIp) = rov  vo*t
= [ (Rocs(POIo)*sinOo)^2 + (Rocs(POIo)*cosOo  vos*t)^2 ]^(1/2)
Vonv(PART) X rph = vos*sinOp*Pph
Vonv(PART) X rph = vos*sinOp*Pph
Vonv(PART) X EIodv(POIp(t),t) = vos*sinOp*EIodv(POIp(t),t)*Pph
rps*sinOp = Rocs(POIo)*sinOo
/* Substituting for crossproducts : This seems WRONG > should be Bi(rov,t)?
/$ Bi(r  v*t,t)
/%^% BIodv(ro  vo*t,t)
/*what follows is a "normal interpretation" founded on fixed geometry
/$ B(r´,t´) = q/c*(vr´)/r´s^2 + v/cEi(r´,t´)
/%^% B(rov,t) = Q(particle)/c*(Vonv(PART)Rpcv(POIp))/rps^2 + Vonv(PART)/cEIodv(Rpcv(POIp(t),t)
/*(NOTE: rov NOT r´!!)
=> first term is a constant, so derivative is zero.
Note that an expression is needed for what the observer POI "sees"
and that, is a dynamic (differential) expression
/$ B(r´,t´) = q/c*(vB(r,t) rr´)/r´s^2 + v/cEi(r´,t´)
/* (NOTE: rov NOT r´!!)
1st term > substitute for [sinOp,rps]
/$ ∂[∂(t): q/c*vs*sin(θ´)/r´s^2*φ´hat
= vs/c*q*(Rocs(POIo)*sin(θ)/r´s)/r´s^2*φ´hat
= vs/c*Rocs(POIo)*sin(θ)*q/r´s^3*φ´hat
2) = vs/c*Rocs(POIo)*sin(θ)*q/r  v*t^3*φ´hat
/* 2nd term > substitute for sinOp
/$ 1/c*vs*sin(θ´)*Ei(r  v*t,t)*φ´hat
= vs/c*(Rocs(POIo)*sin(θ)/r´s)*Ei(r  v*t,t)*φ´hat
3) = vs/c*Rocs(POIo)*sin(θ)/r  v*t*Ei(r  v*t,t)*φ´hat
/*substituting (2),(3) into (1) :
/$4) Bi(r  v*t,t)
= vs/c*Rocs(POIo)*sin(θ)*q/r  v*t^3*φ´hat
+ vs/c*Rocs(POIo)*sin(θ) /r  v*t*Ei(r  v*t,t)*φ´hat
= vs/c*Rocs(POIo)/r  v*t*sin(θ)*φ´hat
*[ q/r  v*t^2 + Ei(r  v*t,t) ]
/* NOTE : If I simply leave Op in expression with Ei(ro  vo*t,t)
I get :
/$5) Bi(r  v*t,t)
= vs/c*φ´hat
* [ q*sin(θ)*Rocs(POIo)/r  v*t^3
+ sin(θ´)*Ei(r  v*t,t) ]
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Lucas_Typo_or_omission
04_15rev3
E,B for symmetry point charge @v_const
Adapted non[derivative, integral] form of Amperes law
/$ Bi(r  v*t,t)
= vs/c*φ´hat
* [ q*sin(θ)*Rocs(POIo)/r  v*t^3
+ sin(θ) *Ei(r  v*t,t) ]
/%^% BIodv(POIo,t)
= Vons(PART)/c*APpch
* [ Q(particle)*sin(Aθoc(POIo))*Rocs(POIo)/Rpcs(POIo(t),t)^3
+ sin(Aθpc(POIo(t),t)) *EIods(POIo,t) ]
/* ( 09Jan2016 WRONG  I have Op rather than Oo in 2nd term.
NOTE : If I repace Op in expression with Ei(ro  vo*t,t)
Difference between static and induced fields!?!
I havent wrapped my head around the details.
 must recheck basis of angles!
/*_endCmd
/**********************************************************
/*> (416) E,B for symmetry point charge @v_const
/*/*$ cat >>"$p_augmented" "$d_augment""04_16 work.txt"
First try : ?date?., third attempt 06Sep2015, rev1 24Sep2015,
09Jan2016 rev4Lucas415, 22May2016 used my expression from file
"Howell  Background math for Lucas Universal Force, Chapter 4.odt" HFLN format :
# Start with (415) Howell's version
E,B for symmetry point charge @v_const
Adapted non[derivative, integral] form of Amperes law
/$ Bi(ro  vo*t,t)
= vos/c*Pph
* [ q*sinOo*ros/rov  vo*t^3
+ sinOp*Ei(ro  vo*t,t) ]
/* taking the ∂/∂t of (415) Howell's
/$1) dp[dt : Bi(ro  vo*t,t)]
= vos/c*Pph
* [ dp[dt : q*sinOo*ros/ro  vo*t^3 ]
+ dp[dt : sinOp*Ei(ro  vo*t,t) ] ]
/*
yy)"Pseudoconstant variables/expressions" within partial derivatives wrt time
yy)Remove : [c,L(v),v,b,q,rp,ro  vo*t^(n),Op,sinOp,Pp]  Chap4 v=constant, Chap5 v=variable.
yy)Retain : [t,E,B,ro,Oo,sinOo] are the key variables to retain within derivatives
for Chapter 4 verifications
# yielding
/$2) dp[dt : Bi(ro  vo*t,t)]
= vos/c*Pph
* [ q /ro  vo*t^3 *dp[dt : ros*sinOo ]
+ sinOp *dp[dt : Ei(ro  vo*t,t)] ]
/* From "Howell  Key math info & derivations for Lucas Universal Force.odt"
Section II.4 :
/$II.4.9) dp[dt : ros*sinOo] = 0
/*OOOPPPS!!!  q term drops out as it is a constant
and the derivative is zero
/$ dp[dt : Bi(ro  vo*t,t)]
= vos/c*Pph
* [ q /ro  vo*t^3 *dp[dt : ros*sinOo ]
+ sinOp *dp[dt : Ei(ro  vo*t,t)] ]
/*10Jan2016 The problem here is that in (414) and (415) the "q" expressions are constant  leading to a zero derivative.
dp[dt : ros*sinOo] = vos*sinOo*(cosOo  1)
/*therefore :
/$4) dp[dt : Bi(ro  vo*t,t)]
= vos/c*Pph
* [ q/ro  vo*t^3 *vos*sinOo*(cosOo  1)
+ sinOp*dp[dt : Ei(ro  vo*t,t) ] ]
= vos/c*Pph
* [ q*vos*sinOo*(cosOo  1)/ro  vo*t^3
+ sinOp*dp[dt : Ei(ro  vo*t,t) ] ]
/* This is very different from Lucas, at least in appearance.
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Possible_Lucas_error_or_omission
04_16rev4Lucas415
E,B for symmetry point charge @v_const
Derivative form of Amperes law
/$ ∂[∂(t): Bi(r  v*t,t)]
= vs/c*Rocs(POIo)*sin(θ)*φ´hat*
[ 3*q*vs/c*(Rocs(POIo)*cos(θ)  vs*t)/r  v*t^5
+ 1/Rocs(POIo)/c*∂[∂(t): Ei(r  v*t,t)] ]
/% dp[dt : Bi(ro  vo*t,t)]
= vos/c*Pph
* [ q*vos*sinOo*(cosOo  1)/ro  vo*t^3
+ sinOp *dp[dt : Ei(ro  vo*t,t)] ]
/* Note how different my version is ??!!??
/% ∂[∂(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(particle) /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))
}
/* 10Jan2016 OOOPPPSSS  qterm is zero (constant has zero derivative)
09Jan2016 WRONG! : I use Op not Oo for 2nd term RHS (as with (415), different terms
This could partially be due to differentiation, but not all.
??It seems to me that Lucas has dropped the "Vons(PART)*cos(AOpc(POIo,t))*EIpds(POIo,t)/Rpcs(POIo(t),t)" term.
But there should be an EIpds(POIo,t) term!
/* It seems clear to me that Lucas has :
 dropped the "Vons(PART)*cos(AOpc(POIo,t))*EIpds(POIo,t)/Rpcs(POIo(t),t)" term. But there should be an EIpds(POIo,t) term!
22Aug2019 NO!??? I factored out Vons(PART)*cos(AOpc(POIo,t)) ???
 has too many "c"s
22Aug2019  do dimensional analysis!!
 has the term "(ros*cosOo  vos*t)/ro  vo*t" in the first term in parenthesis.
 I have NO real idea of why the (ros*cosOo  vos*t) pops up anyways, and why the cos term all of a sudden (spherical coords does not explain this!)
30Mar2018 do spherical coordinate derivative!?!!
NOTE!!! : use "/media/bill/SWAPPER/Qnial/MY_NDFS/matrix derivatives in [cartesian, cylindrical, spherical] coordinates.ndf"
/*_endCmd
/**********************************************************
/*> (417) Spherical coordinate transforms
Howells clarifications  Lucas converts to spherical coordinates
see Howell  Symbols for Bill Lucas's book "The Universal Force, vol1"
Special figure for this
Expression for rps(Rocs,Oo,vo,t)
From Diagram :
/$ r = Rocs = [ (Rocs*sin(θ))^2 + (Rocs*cos(θ))^2 ]^(1/2)
1) r´ = r´s = [ (r´s*sin(θ´))^2 + (r´s*cos(θ´))^2 ]^(1/2)
/* Noting that :
/$2) r´s *sin(θ´) = Rocs*sin(θ)
3) Rocs*cos(θ) = vs*t + r´s*cos(θ´)
/* Substituting 2) & 3) into 1) :
/$417a) r´s = [ (Rocs*sin(θ))^2 + (Rocs*cos(θ)  vs*t)^2 ]^(1/2)
/* Expression for vov X r´
/$5) v X r´ = v X (r  v*t) = v X r  v X v*t
but v X v = 0 (collinear) so :
6) v X r´ = v X r = v*r*sin(θ)*φ´hat
417b) v X r´ = vs*Rocs*sin(θ)*φ´hat
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Re_check_later
04_17rev2
Spherical coordinate transforms
/$ r´ = r  v*t = sqrt((Rocs*sin(θ))^2 + (Rocs*cos(θ)  vs*t)^2)
vr´ = v(r  v*t) = vr = vs*Rocs*sin(θ)*φ´hat
/% Rpcs(POIp) = { [Rocs(POIo,t)*sin(Aθoc(POIo))]^2 + [Rocs(POIo,t)*cos(Aθoc(POIo)  Vons(PART)*t]^2 }^(1/2)
????Vonv(PART) X Rpcv(POIp) = Vons(PART)*Rocs(POIo,t)*sin(Aθoc(POIo))*APph ???? *Rodh(Vonv_X_Rpcv(POIo))
/* ( 02Jan2016 OK  easy BUT, I still need to check angle basis below...
Recheck my HFLN expression!
/**********************************************************
/*> (418) Changing magnetic flux linked by a circuit proportional to induced E field around the circuit
from Jackson 1999, p211h0.45 Eqn (5.141)
/$ ∮[•d(l´),.over.C: ET) = 1/c*∂[∂t: ∮[dArea,.over.S: (B•n))
/% ∮[•d(l´),.over.C: E) = 1/c*∂[∂t: ∮[∂(Area),.over.S: (B•n))
/*where :
nh is a unit vector normal to the surface S at the point of integration
from Jackson to Lucas notation
 n = nh
 B = Bi(ro  vo*t,t) (observer frame),
 E = Eip(r´,t´) (particle/system frame)
/$ ∮[•d(l´),.over.C: Ei´(r´,t´)) = 1/c*∂[∂t: ∮[dArea,.over.S: (Bi(r  v*t,t)•nh))
/%^% ∮[•d(l´),.over.C: Eip(Rpcv(POIp),t´)) = 1/c*∂[∂t: ∮[dArea,.over.S: (BIodv(ro  vo*t,t)•Roch(POIo)))
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Question
04_18
Faradays law > B linked by circuit = induced E around circuit
/$ ∮[•d(l´),.over.C: Ei´(r´,t´)) = 1/c*∂[∂t: ∮[dAreap,.over.S: Bi(r  v*t,t)•nh)
∮[•d(l´),.over.C: Ei´(r´,t´)) =  *∂[∂t: ∫[dArea,.over.S: Bi(r  v*t,t)•nh)
/%^% ∮[•d(l´),.over.C: (EIpdv(POIp) = 0)) = 1/c*∂[∂t: ∮[dArea,.over.S: (BIodv(POIo,t) •Roch(POIo)))
∮[•d(l´),.over.C: Eip(Rpcv(POIp),t´)) =  *∂[∂t: ∫[dArea, .over.S: BIodv(ro  vo*t,t)•Roch(POIo))
/* Perfect  simple formula translation, BUT is Ep = E, as it looks
to me that these are used differently/simultaneously p68&69, but it is NOT explained! What is it?
/**********************************************************
/*> (419) E,B for symmetry point charge @v_const  Stokes theorem
/*/*$ cat >>"$p_augmented" "$d_augment""04_19 work.txt"
Faradays law (42) can be rewritten using Stokes theorem
Lucas (402) := Faradays_Law
/$ ∫E(Rpcv,t´)•∂(l) = 1/c*∂[∂(t): ∫B(r,t)•nda]
/*very similar to (418)?
Stokes theorem  Kreyszig1972 p364h0.6 Eqn (8.101)
/$ ∬[∂(Area): (curl(v))_n) = ∮[ds,.over.C: v*t)
/* where (curl v)_n = (curl v)•n is the component of curl v in the direction of a unit normal vector n of S; the integration around C is taken in the sense shown in Figure 166, and vt is the component of v in the direction of the tanjent vector of C in Fig 166 (Howell : righthand curl!)
Faradays law as expressed in (42)≈ RHS of (418), so this is OK with a warning about the switched ordering of differentiation and integration
BUT its not clear what has been done on the LHS :
stated as :
/$ (Ei´(r´,t´)  1/c*vBi(r´,t´))
/*BUT  this MUST have ∂[∂t: of B (differential or integral)
My GUESS is that E0 is needed as per (413)
/$ B(r´,t´) = (v/c)[ E0(r´,t´) + Ei(r´,t´) ]
/* NOTE : units dont work for (22) versus (23), and (419) appears to have the same problem.
Replacing Ei(r´,t´) with Ei´(Rpcv,t´)  E0(r´,t´)
/$1) ∫[Ei´(Rpcv,t´)  E0(r´,t´))•∂(l) = 1/c*∂[∂(t): ∫B(r,t)•nda]
/* failedtry :
For E0(r´,t´), for a point charge : see Jackson1999 p28h0.1
/$(1.7) ET•n*∂(Area) = q/4/π/ε0*cos(θ)/r^2*∂(Area)
ET•n = q/4/π/ε0*cos(θ)/r^2
/* or spherical surfaces centered on a particle
/$ ET = q/4/π/ε0/r^2
(1.8) ET•n*∂(Area) = q/4/π/ε0*dΩ
/* where dΩ = solid angle subtended by da at the position of the charge
/$ r^2*dΩ = cos(θ)*∂(Area)
/* Taking E0(r´,t´) = E in 1.7 = q/4/π/ε0/r^2, and substituting
Nyet
Oops
take Amperes Law (41)!!!
Lucas04_01 := Generalized_Amperes_Law, note change of refFrame
/$ B(r´,t´) = v/c × E0(Rpcv,t´)
/* so
/$2) E0(Rpcv,t´) = c/v*B(r´,t´)
/*OOPS!!! CROSS PRODUCT!
putting this into 1 above
/$3) ∫[E´(Rpcv,t´)  c/v*B(r´,t´))•∂(l) = 1/c*∂[∂(t): ∫B(r,t)•nda]
/*write out more specifically (Howell notation) :
/$4) ∮[•d(l´),.over.L: (E´(Rpcv,t´)  c/v*B(r´,t´)))
= 1/c*∂[∂(t): ∫[dArea,.over.A: B(r,t)•n)]
/*Notice that I didn't have to apply Stokes theorem!
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Question
04_19
( E,B for symmetry point charge @v_const
add Faradays law to [Ampere]
/$Lucas ∮[•d(l´),.over.L: (Ei´(r´,t´)  v/cBi(r´,t´))) = 1/c* ∮[dAreap,.over.Ap: (∂[∂(t): Bi(r  v*t,t)]•n)
Howell ∮[•d(l´),.over.L: (Ep(r´,t´)  c/v*Bi(r,t))) = 1/c*∂[∂(t): ∮[dArea,.over.A: B(r,t]•n)
/%^%Lucas ∮[•d(l´),.over.closedcurveL: EIpdv(POIp)  Vonv(PART)/c*BIodv(POIp(t),t)) = 1/c*∮[dAreap,.over.Ap: (∂[∂(t): BIodv(POIo,t)]•n}
Howell ∮[•d(l´),.over.closedcurveL: EIpdv(POIp)  c/Vonv(PART)*BIodv(POIp(t),t)) = 1/c *∂[∂(t): ∮[*∂(Area),.over.A: BIodv(POIo,t)]•n}
/* better to use ∬ rather than ∮ for Bi.over.A ??
PROBLEMS  crossproduct correction to my deriv, v/cBi(r´,t´ versus c/v*Bi(r,t),
changed order of diff/integ B(r,t) vs Bi(ro  vo*t,t),
/*_endCmd
/**********************************************************
/*> (420) Convective_derivative
Jackson1999 p210h0.9 footnote "*" convective derivative :
/$ ∂/∂(t) = ∂[∂t: + v•∇
/* where nabla ∇p = gradient in particle refFrame =
Cartesian : d(dx : ?)*xvh + ...
Spherical : ....
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
04_20
Convective_derivative
/$ ∂/∂(t) = ∂[∂t: + v•∇p]
/% d/∂(t) = ∂[∂t: + v•∇p]
/* OK  straight defn. Usage explained by Jackson (to generalize Faradays Law?)
HOWEVER  I should have been using convective derivatives elsewhere!!  eg as I did with dp[dt : E] in Background
(eg file "Howell  Background math for Lucas Universal Force, Chapter 4.odt")
/**********************************************************
/*> (421) convective derivative of Total magnetic flux density Bi
/*/*$ cat >>"$p_augmented" "$d_augment""04_21 work.txt"
Lucas's derivation :
/$ ∂[∂t: Bi(r  v*t,t))
= ∂[∂(t): Bi(r  v*t,t)] + (v•∇´)*Bi(r  v*t,t)
= ∂[∂(t): Bi(r  v*t,t)] + ∇´ [Bi(r  v*t,t)v] + v* [∇´•Bi(r  v*t,t)]
= ∂[∂(t): Bi(r  v*t,t)] + ∇´ [Bi(r  v*t,t)v]
/* since ∇´•Bi = 0
Irrelevant :
04_19 Point charge with symmetry @ v_const, add Faradays law (Ampere!)
/$ ∮[•d(l´),.over.L: (Ei´(r´,t´)  v/cBi(r´,t´))) = 1/c* ∫[dAreap,.over.Ap: (∂[∂(t): Bi(r  v*t,t)]•n)
/* 04_20 Convective_derivative
/$ ∂/∂(t) = ∂[∂t: + v•∇´
/* Apply Lucas's (420) to Lucas's (419)
/$ (∂[∂t: + v•∇´))
/* First try  Howells derivation
Apply Lucas's (420) convDeriv to Bi(ro  vo*t,t) B total, observer frame)
/$ ∂/∂(t)[Bi(r  v*t,t)] = (∂[∂t: + v•∇´)[Bi(r  v*t,t)]
/$ ∂/∂(t)[Bi(r  v*t,t)] = ∂[∂(t): Bi(r  v*t,t)] + (v•∇´)[Bi(r  v*t,t)]
/* Vector operations from Jackson1999 inside front cover
/$ ∇(a•β) = (a•∇)*β + (β•∇)*a + a(∇β) + β(∇a)
/* therefore
/$ (a•∇)β = ∇(a•β)  [+ (β•∇)*a + a(∇β) + β(∇a) ]
/* HOWEVER  I dont fully comprehend the form (a•∇´)b, presumably eg
/$ (a•∇) = (a1*n1 + a2*n2 + a3*n3) • (∂/∂x1*n1 + ∂/∂x2*n2 + ∂/∂x3*n3)
(a•∇) = (a1*∂/∂x1 + a2*∂/∂x2 + a3*∂/∂x3)
/* therefore
/$ (a•∇)β = (a1*∂/∂x1 + a2*∂/∂x2 + a3*∂/∂x3) * β
/* but what does that "*b" mean?  assume "kind of" a dot product ???
/$ (a•∇)β = a1*∂/∂x1(b1) + a2*∂/∂x2(b2) + a3*∂/∂x3(b3)
/* take
/$ a = v, β
= Bi(r  v*t,t), ∇ = ∇´
(v•∇´)* [Bi(r  v*t,t)]
= + ∇(v• Bi(r  v*t,t))
 [ + (Bi(r  v*t,t)•∇)*v
+ v(∇ Bi(r  v*t,t))
+ Bi(r  v*t,t)(∇v)
]
/* Oops, wrong one
/$ ∇(aβ) = a*(∇•β)  β*(∇•a) + (β•∇)*a  (a•∇)*β
/* therefore
/$ (a•∇)β = a*(∇•β)  β*(∇•a) + (β•∇)*a  ∇(aβ)
/* again, take a = v, b = Bi(ro  vo*t,t), ∇ = ∇´
/$ (v•∇´) Bi(r  v*t,t)
= + v(∇´•Bi(r  v*t,t))
 Bi(r  v*t,t)(∇´•v)
+ (Bi(r  v*t,t)•∇´)v
 ∇´(v Bi(r  v*t,t))
/* Using ∇´•B = 0, ∇´v = 0 for constant relative v
/$ (v•∇´)* Bi(r  v*t,t)
=  Bi(r  v*t,t)(∇´•v)
 ∇´(vBi(r  v*t,t))
/* Summarizing Howells development :
/$ ∂/∂(t)[Bi(r  v*t,t)] = ∂[∂(t): Bi(r  v*t,t)] + (v•∇´)[Bi(r  v*t,t)]
∂/∂(t)[Bi(r  v*t,t)]
= ∂[∂(t): Bi(r  v*t,t)]
 Bi(r  v*t,t)(∇´•v)
/* rearrange
/$ ∂/∂(t)[Bi(r  v*t,t)] = ∂[∂(t): Bi(r  v*t,t)]  ∇´(vBi(r  v*t,t))  Bi(r  v*t,t)(∇´•v)
Can I combine? :  ∇´(vBi(r  v*t,t))  Bi(r  v*t,t)(∇´•v)
From Kreyszig
/$ aβ =  βa
/* Try
/$ β(c∂) = (β∂)c  (βc)∂ on
∇´(vBi(r  v*t,t)) = (∇´.Bi(r  v*t,t))*v  (∇´.v)*Bi(r  v*t,t)
/* but
/$ ∇´.Bi(r  v*t,t) = 0
/* so
/$ ∇´(vBi(r  v*t,t)) = 0  (∇´.v)*Bi(r  v*t,t)
/* putting this in
/$  ∇´(v Bi(r  v*t,t))  Bi(r  v*t,t)*(∇´•v)
=  [ (∇´.v)* Bi(r  v*t,t)]  Bi(r  v*t,t)*(∇´•v)
= 0
/* OOPS  that wiped me out! should check why some day, wheres my mistake?
2nd try  Howells derivation
Apply Lucas's (420) convDeriv to Bi(ro  vo*t,t) B total, observer frame)
/$ ∂/∂(t)[Bi(r  v*t,t)] = (∂[∂t: + v•∇´)[Bi(r  v*t,t)]
∂/∂(t)[Bi(r  v*t,t)] = ∂[∂(t): Bi(r  v*t,t)] + (v•∇´)[Bi(r  v*t,t)]
/* scalar triple products Kreyszig Section 5.9 p213216
/$ β(c∂) = (β∂)*c  (βc)*∂
(β∂))*c = β(c∂) + (βc)*∂
/* again, take b = ∇´, c = Bi(ro  vo*t,t), d = v
(∇´v)Bi(ro  vo*t,t) = ∇´(Bi(ro  vo*t,t)v) + (∇´Bi(ro  vo*t,t))*v so
/* OK  this looks closer?
/$ Given ∇´Bi(r  v*t,t) = 0 then
(∇´v)Bi(r  v*t,t) = ∇´(Bi(r  v*t,t)v) + 0
/* such that
/$ ∂/∂(t)[Bi(r  v*t,t)] = ∂[∂(t): Bi(r  v*t,t)] + ∇´[Bi(r  v*t,t)v]
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
04_21
convective derivative of Total magnetic flux density Bi
/$ ∂/∂(t)[Bi(r  v*t,t)] = ∂[∂(t): Bi(r  v*t,t)] + ∇´[Bi(r  v*t,t)v]
∂/∂(t)[Bi(r  v*t,t)] = ∂[∂(t): Bi(r  v*t,t)] + ∇´[Bi(r  v*t,t)v]
/%^% d/∂(t)[BIodv(ro  vo*t,t)] = ∂[∂(t): BIodv(ro  vo*t,t)] + ∇´[BIodv(ro  vo*t,t)Vonv(PART)]
dT[∂(t): BIodv(POIo,t)] = ∂[∂(t): BIodv(POIo,t)] + ∇´[BIodv(POIo,t)Vons(PART)]
/* OK  2nd try gets same as Lucas
HOWEVER  I should have been using convective derivatives elsewhere!!  eg as I did with dp[dt : E]
(eg file "Howell  Background math for Lucas Universal Force, Chapter 4.odt")
/*_endCmd
/**********************************************************
/*> (422) KelvinStokes integration of convective derivative of Bi total
p69h0.2 apply KelvinStokes theorem to Lucas04_21 : gives Lucas04_22
https://en.wikipedia.org/wiki/Kelvin%E2%80%93Stokes_theorem
Let γ: [a, b] → R2 be a Piecewise smooth Jordan plane curve. The Jordan
curve theorem implies that γ divides R2 into two components, a compact
one and another that is noncompact. Let D denote the compact part that
is bounded by γ and suppose ψ: D → R3 is smooth, with S := ψ(D).
If Γ is the space curve defined by Γ(t) = ψ(γ(t))[note 1] and F is a
smooth vector field on R3, then:
/$ ∮(dΓ: F) = ∬[∂(Area): ∇F)
/* Note 1 : γ and Γ are both loops, however, Γ is not necessarily a Jordan curve
STRANGE : I dont see the point of deriving (422) at all!??
I have simply applied KelvinStokes to (419) in the next section (423)
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Howell_incomplete
04_22
(KelvinStokes integration of convective derivative of Bi total
Lucas ??  Integral form of Faradays_Law_for_moving_circuit  magnetic fields???
/$ 1/c*∮*∂(Area)´(∇´[Bi(r  v*t,t)v]•np) = 1/c*∮•d(l´)([Bi(r  v*t,t)v])
/* STRANGE : I dont see the point of deriving 422 at all!??
didnt do  I dont see the point of deriving 422 !?!
I have simply applied KelvinStokes to 419 in the next section 423
/**********************************************************
/*> (423) Faradays_Law_for_rest_circuit integral form E,B
/*/*$ cat >>"$p_augmented" "$d_augment""04_23 work.txt"
Note : is the prime following n in Lucas (423) an error?
Howell WRONG > Also  falls immediately from (419) with v = 0,
but then r  vt = r as well
1. Start from precendents
04_19 E,B for symmetry point charge @v_const, add Faradays law to [Ampere]
/$ ∮[•d(l´),.over.L: (Ei´(r´,t´)  v/cBi(r´,t´)))
=  1/c* ∮[dAreap,.over.Ap: (∂[∂(t): Bi(r  v*t,t)]•n)
/* rebase LHS of (419)
/$ ∮[•d(l´),.over.L: (Ei´(r  v*t,t)  v/cBi(r  v*t,t)])
=  1/c* ∮[dAreap,.over.Ap: (∂[∂(t): Bi(r  v*t,t)]•n)
1) ∮[•d(l´),.over.L: (Ei´(r  v*t,t))
= + ∮[•d(l´),.over.L: v/cBi(r  v*t,t))
 1/c*∮[dAreap,.over.Ap: (∂[∂(t): Bi(r  v*t,t)]•n)
/* 2. Howell application of KelvinStokes to 1st term RHS
Let Γ = lp, dS = da
/$ ∮ (•d(l´),.over.L : v/cBi(r  v*t,t))
= ∬[∂(Area): ∇(v/cBi(r  v*t,t)))
/* for RHS
/$ ∇(v/cBi(r  v*t,t))
a) Vector operations from Jackson1999 inside front cover
/$ ∇(aβ) = a(∇β)  β(∇a) + (β∇)a  (a∇)β
let a=v β=Bi(r  v*t,t)
∇(v/cBi(r  v*t,t))
= + v(∇Bi(r  v*t,t))
 Bi(r  v*t,t)*(∇v)
+ (Bi(r  v*t,t)∇)*v
 (v∇)β
/*b) scalar triple products Kreyszig1972 Section 5.9 p213216
/$ β(c∂) = (β∂)c  (βc)∂
take β = ∇´, c = Bi(r  v*t,t), ∂ = v
∇´ (Bi(r  v*t,t)v)
= + (∇´v)*Bi(r  v*t,t)
 (∇´ Bi(r  v*t,t))v
for constant v, ∇´v=∇´B=0, so
c) ∇´(Bi(r  v*t,t)v) = 0
/* Using (c) in (1)
/$1) ∮[•d(l´),.over.L: (Ei´(r  v*t,t))
= + ∮[•d(l´),.over.L: v/cBi(r  v*t,t))
 1/c*∮[dAreap,.over.Ap: (∂[∂(t): Bi(r  v*t,t)]•n)
/* Result
/$ ∮[•d(l´),.over.L: Ei´(r  v*t,t)) =  1/c*∮[dAreap,.over.Ap: (∂[∂(t): Bi(r  v*t,t)]•n)
/* key question is whether ?more rigorous? Jackson expression would yield
the same result as my simple Kreyszig application?
(maybe check later  in (419) this seemed to be a problem?)
04_21 convective derivative of Total magnetic flux density Bi
/$ ∂/∂(t)[Bi(r  v*t,t)] = ∂[∂(t): Bi(r  v*t,t)] + ∇´[Bi(r  v*t,t)v]
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Re_check_later
04_23
Faradays_Law_for_rest_circuit integral form E,B
/$ ∮[•d(l´): Ei(r  v*t,t)) = 1/c*∮[dAreap: ∂[∂(t): Bi(r  v*t,t)]•np)
/%^% ∮[•d(l´): EIodv(POIo,t)) = 1/c*∮[dAreap,.over.Ap: ∂[∂(t): BIodv(POIo,t)]•R_A_PI2_odh)
/* OK  seems good, Note that this is for v=0, but how is this different than (42) Faradays Law?
some worry about vector formulae Jackson1999 versus Kreyszig1972
Note : is the prime following n in Lucas04_23 an error?
where : Rodh(Vonv_X_Rpcv(POIo)) is the unit normal vector at each point on area A
/*_endCmd
/**********************************************************
/*> (424) E&B for [Faradays + part/obs frameTrans]  towards FU_Faradays_Law
/*/*$ cat >>"$p_augmented" "$d_augment""04_24 work.txt"
p69h0.5 Galilean invariance requires that in this case
(circuit at rest v=0, Lucas04_23) Ep = E and Fp = F,
1. Starting point, 11Sep2015
04_12 E transformation particle to observer frames
/$ ET(r´,t´) = ET(r  v*t,t)
/* 04_16Derivative form of Ampere's law
Take early form
/$ 1/c*∂[∂(t): Bi(r  v*t,t)]
= v/c*r*sin(θ)*φ´* [ 3*q*v/c*(r*cos(θ)  v*t)/r  v*t^5
+ 1/r/c*∂[∂(t): Ei(r  v*t,t)]
]
/* doesn't make sense
2. try (412)&(419) rather than (412)&(416)
04_19E,B for symmetry point charge @v_const . add Faradays law to [Ampere]
/$ ∮[•d(l´),.over.L: (Ei´(r´,t´)  v/cBi(r´,t´)))
= 1/c*∮[dAreap,.over.Ap: (∂[∂(t): Bi(r  v*t,t)]•n)
/* doesn't make sense,
3. try (411)&(413) rather than (412)&(416)
from 04_11 E&B_fields_static_plus_induced
/$ ET(r,v,t) = E0(r,t) + Ei(r,v,t) B(r,v,t) = B0(r,t) + Bi(r,v,t)
/* given Ep = E
/$ Ei´(r´,v´,t´) = Ei(r,v,t)  E0(r,t)
/* from 04_13 Total B magnetic flux density as induced from E0 + Ei
/$ B(r´,t´) = (v/c)[ E0(r´,t´) + Ei(r´,t´) ]
/* reexpress in observer frame
/$ B(r  v*t,t) = (v/c)[ E0(r  v*t,t) + Ei(r  v*t,t) ]
/* close, but (v/c) in wrong place...
4.**what is the "inverse crossproduct (_inv)" function such that
/$ a=βc > _inv(β,c) = c [like a=β*c > /(a,β)=c ]
/* 5. **first look at dot product > inverse dot product
/$ a=β•c > •_inv(a,β) = c
a=β•c = b1*c1 + b2*c2 + b3*c3
c = (β•c) * [n1*c1/(β•c) + n2*c2/(β•c) + n3*c3/(β•c)]
= a * [n1*c1/ a + n2*c2/ a + n3*c3/ a ]
/* unfortunately, this requires explicit knowledge of c
Hold this for now (interesting challenge  but MAYBE underconstrained?)
6. Try dotmultiplying by vector to simplify
/$ B(r  v*t,t) = (v/c)[ E0(r  v*t,t) + Ei(r  v*t,t) ]
(v/c)B(r  v*t,t) = (v/c)( (v/c)[ E0(r  v*t,t) + Ei(r  v*t,t) ] )
using pattern a(βc) = (aβ)c
(v/c)B(r  v*t,t) = (v/c) (v/c) ( E0(r  v*t,t) + Ei(r  v*t,t) )
/* but (v/c)(v/c) = 0 as a vector is collinear with itself!?!
strange "wrong answer" of very basic nature  what am I doing wrong
with vector operations?
Interesting challenge, but taking too much time and Im stuck.
12Sep2015 08h37  Ill come back to this later
23May2016 try multiplying by vX rather than v
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Possible_Lucas_error_or_omission
04_24
E&B for [Faradays + part/obs frameTrans]  towards Fu_Faradays_Law
/$ Ei(r´,v´,t´) = Ei(r  v*t,t) + 1/c*[vBi(r  v*t,t)]
/%^% EIpdv(POIo(t),t) = EIodv(POIo,t) + 1/c*[Vonv(PART)BIodv(POIo,t)]
/* !!!WRONG  On hold as I havent been able to "move" (v/c) to the right place..
AND I still must show that Ep = E and Fp = F !!
/*_endCmd
/**********************************************************
/*> (425) E&B for [Faradays + part/obs frameTrans]  towards FU_Faradays_Law
Lucas development Derivation of the Lorentz Force
/$ Fp(r´,t´)
= F0(r´,t´) + Fi(r´,t´)
= q*E0(r´,t´) + q*Ei(r´,t´)
= q*E0(r  v*t,t) + q*Ei(r  v*t,t) + q/c*[vBi(r  v*t,t)]
/* 1. start with
/$ Fp(r´,t´)
= F0(r´,t´) + Fi(r´,t´)
= q*E0(r´,t´) + q*Ei(r´,t´)
/* This is nice, but F = q*E is NOT stated in the original axioms, even though it is basic, so perhaps assumed.
Utilising (424), plus putting in the observer frame (r´,t´) > (ro  vo*t,t)
(note the change from Ei(r´,t´) to Ei(r´,v´,t´) )
/$ Fp(r´,t´) = q*E0(r  v*t,t´) + q*( Ei(r  v*t,t) + v/cBi(r  v*t,t))
/%^ Fp(Rpcv(POIp),t´) = Q(particle)*E0odv(Rpcv(POIp),t´) + Q(particle)*( EIodv(POIp(t),t) + Vonv(PART)/cBIodv(POIo,t))
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
04_25
Derivation of the Lorentz Force
/$ Fp(r´,t´) = q*E0(r  v*t,t) + q*Ei(r  v*t,t) + q/c*[vBi(r  v*t,t)]
/% FTpdv(POIo(p),t) = Q(particle)*{ E0odv(POIo,t) + EIodv(POIo,t) + Vons(PART)/cBIodv(POIo,t) }
/* OK  straightforward from 424
12Sep2015  I still have problems with 424
/**********************************************************
/*> (426) Derived Lorentz Force F_L(const v : q,E,Bi)
starting with (425)
04_25 Derivation of the Lorentz Force
/% FTpdv(POIo,t) = Q(particle)*{ E0odv(POIo,t) + EIodv(POIo,t) + Vons(PART)/cBIodv(POIo,t) }
/* from (45a) given that
/$ E0(r  v*t,t) + q*Ei(r  v*t,t)
Fp(r´,t´) = q*ET(r  v*t,t) + q/c*[vBi(r  v*t,t)]
/* p69h0.8 "... Note that the Lorentz force law has been derived from Galilean
invariance and the experimental fact that the fields are a physical extension
of the charge making the electromagnetic force a contact type force!
The Lorentz Force law should no longer be considered a fundamental axiom of
electrodynamics. ..."
25Aug2015 Howell  This is an important clarification of what Lucas means
by the expression "contact force". I have not considered "fields" to be a
contact force at all, and others may be confused by this terminology as well.
Perhaps this is to contrast his fieldbased approach to the boson "force
particles" of modern physics?
from http://particleadventure.org/fermibos.html
A fermion is any particle that has an odd halfinteger (like 1/2, 3/2, and
so forth) spin.
Quarks and leptons, as well as most composite particles, like protons and
neutrons, are fermions.
Bosons are those particles which have an integer spin (0, 1, 2...).
All the force carrier particles are bosons, as are those composite particles
with an even number of fermion particles (like mesons)
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
04_26
Derived Lorentz Force, F_L(const v : q,E,Bi)
/$ F (r´,t´) = q*ET(r  v*t,t) + q/c*[vBi(r  v*t,t)]
/% F_LORENTZpdv(POIo,t) = Q(particle)*{ EIodv(POIo,t) + Vons(PART)/cBIodv(POIo,t) }
/* OK  very straightforward
/**********************************************************
/*> (427) Lorentz Force
/*/*$ cat >>"$p_augmented" "$d_augment""04_27 work.txt"
Differential form of Faradays Law (423)
04_23 Faradays_Law_for_rest_circuit integral form E,B
/$ ∮[•∂(l)′: Ei(r  v*t,t))
= 1/c*∮[dArea′: ∂[∂(t): Bi(r  v*t,t)]•n′)
/* This is an interesting one ...
I want to differentiate the LHS by •dl′, the RHS by *da′
This First attempt is a BOTCH, completely unnecessary and ineffective, but
of interest to see what happens
1. First try "reversing the entanglement"
Faradays Law (42) itself creates the "entanglement"
Equate, using solid angle &
Ω = nonplanar solid angle
area A subtended by solid angle
Ω = A/r^2 = 2*π*(1  cosΘ) = 4*π*sin^2(Θ/2/s")
??planar radians dθ = ds/r where s = segment of circle??
solid steradian dΩ = dA/r
R = radial distance (or displacement for nonsymmetry) from the
charge q to the circular loop, and circular "cap" area
For a perfect sphere around charge, and considering [E,B] flux through
a circular loop and area at constant distance y from charge
(Note  this DOESNT have to be a great circle through the center of
the charge), relating dΩ and dy : (wanda.fiu.edu)
"Figure 2: Geometry of the cross section and the solid angle"
let Ra = the radius of the circular loop defined by [y,Ω]
(vs R = the radius (distance) from the charge to the circular loop )
In this webpage (wanda.fiu.edu) dΩ is defined in terms of a dθ for annular
rings, going from the center of the circular loop to its perimeter :
/$ dΩ_annular = 2*π*R*sinθ*R/R^2*dθ = 2*π*sinθ*dθ
/* 1a. But rather than using this definition (rather than a pie slice of
constant solid angle and planar angle), and [being lazy, wanting to be different,
and not having to deal with radiation problems of spherical symmetry]
(as opposed to constant radiation spherically) Ill use one with
constant dA, dl
/$ Ω_full_sphere = 4*π steradians
A_surface_section = 4*π*Rs^2*Ω_circle/Ω_full_sphere
/* Define θ as angle CCW from some arbitrary starting point on the circular loop
/$ ∂(l) = ((y*dz)^2 + dy^2)^0.5
∂(Area) := 4*π*y^2*dz/
/* Should be the circular area  not "chordal"
/$ y = constant, dy/dz = 0
∂(l) = y*dz
/* AH HAH!  perhaps [θ,φ] as defined here are the sense of Lucas's use
in (415)(417) !!??!!
1b. Wait  go back to the "standard" annular formula, as that provides
a differential solid angle, which in turn is a basis for Area of the
spherical surface "cap" :
/$ Ω_full_sphere = 4*π steradians
dΩ_annular = 2*π*sinθ*dθ
dA_annular = A_sphere*dΩ_cap/Ω_full_sphere
/* and
/$ A_cap = A_sphere*Ω_cap/Ω_full_sphere
∂(l) = ((y*dz)^2 + dy^2)^0.5
∂(Area) := 4*π*y^2*dz/
/* NO WORKEE!  I NEED the other form to relate dl&dA, BUT, this can be
used to calculate A_cap, L_cap
1.b.i
let : θf be the angle from the center of the cap to its
outer (circular) edge
where :
/$ A_cap = A_sphere/Ω_full_sphere*Ω_cap
= 4*π*R^2 / (4*π) * ∫[dθ,0 to θf: 2*π*sinθ*dθ)
∫[dθ,0 to θf: 2*π*sinθ*dθ)
= diff(0 to θf:  2*π*cos(θ))
= 2*π*(cos(θf)  1)
= 2*π*(1  cos(θf))
/* check for θf = π, 2*π*(1  cos(θf)) = 2*π*(1  (1)) = 4*π (correct!)
so :
/$ A_cap = 2*π*(1  cos(θf))*R^2
1.β.ii
/$ let : L_cap = length around outer edge of cap, which is a circle
where : R_cap_edge = R*sinθf
so : L_cap = 2*π*R*sinθf
/* 1.a.i. Going back to "pieslices"
Now, for pieslices of cap, going from φ = 0 to 2*π
From symmetry, I argue that :
/$ : A_pie/A_cap = φ_pie/φ_cap = φ/2/π
so : dA_pie/A_cap = ∂(φ)/2/π
also : L_pie/L_cap = φ_pie/φ_cap = φ/2/π
also : dL_pie/L_cap = ∂(φ)/2/π
/* i.e. same "ratio formulae" for [A_pie, L_pie] and [dA_pie, dL_pie]
which means its easy to jointly integrate?
1.c Reminder of original objective : get the Differential form of
Faradays Law (423)
04_23 Faradays_Law_for_rest_circuit integral form E,B
/$ ∮[•∂(l)′: Ei(r  v*t,t)) = 1/c*∮[dArea′: ∂[∂(t): Bi(r  v*t,t)]•n′)
/* Substituting for dl and da (1.a.i. above using [dL,dA])
/$ ∮[•(L_cap/2/π*∂(φ))′: Ei(r  v*t,t))
= 1/c* ∮[*(A_cap/2/π*∂(φ))′: ∂[∂(t): Bi(r  v*t,t)]•n′)
/* extracting constants
/$ L_cap/2/π *∮[•∂(φ)′: Ei(r  v*t,t))
= 1/c*A_cap/2/π *∮[*∂(φ)′: ∂[∂(t): Bi(r  v*t,t)]•n′)
/* Now both have the same basis of integration. Multiply each side by
2*π and gathering terms :
/$ ∮[•∂(φ)′: Ei(r  v*t,t))
= 1*A_cap/L_cap/c* ∮[*∂(φ)′: ∂[∂(t): Bi(r  v*t,t)]•n′)
where
A_cap/L_cap
= [2*π*(1  cos(θf))*R^2] / [2*π*R*sinθf]
= (1  cos(θf))/sinθf*R
= (1  cos(θf))/sinθf*R
/* Rearranging (I have a dot product "leftover"  I should have used a unit vector
for the dotProd)
/$ 0 = ∮[•∂(φ)′: Ei(r  v*t,t)]
+ ∂[∂(t): Bi(r  v*t,t)]•n′ * (1  cos(θf))/sinθf*R/c
)
/* differentiating both sides by dφ′, (for this [R,θf] are constant) :
/$ 0 = Ei(r  v*t,t)
+ ∂[∂(t): Bi(r  v*t,t)•n′] * (1  cos(θf))/sinθf*R/c
/* finally :
/$ Ei(r  v*t,t) = ∂[∂(t): Bi(r  v*t,t)•n′] *(1)(1  cos(θf))/sinθf*R/c
/* Nuts  1.c was NOT successful (I need to look at it again later)
HOWEVER, my result makes sense in that for the same total B in the cap,
independent of R (Bi with 1/R^2 but area goes up that much),
Ei will go down with 1/R.
2. Do as Lucas suggests for (427) p69h0.8, and convert ONLY the
line integral to a surface integral via Stokes theorem.
Kreyszig1972 p364h0.5 Stkes theorem :
/$ ∬[∂(Area): (∇v)n) = ∮(ds: v*t)
/* from which (using n=n′,∇=∇′) :
/$ ∮[•∂(l)′: Ei(r  v*t,t))
= ∬[∂(Area): (∇′v)n′)
= ∬[∂(Area): (∇′Ei(r  v*t,t))n′)
/* Note that ∇′Ei(r  v*t,t)) is aligned (+,) with Bi
(03Feb2016 as the gradient of E is radially out from the particle!),
and is like the Bi integral in (423). Placing into (423) :
/$ ∬[*∂(Area): (∇′Ei(r  v*t,t))n′) = 1/c*∬[dArea′: ∂[∂(t): Bi(r  v*t,t)]•n′)
/* Differentiating both sides by dA :
/$ (∇′Ei(r  v*t,t))n′ = 1/c*∂[∂(t): Bi(r  v*t,t)]•n′
/* One can remove the common •n′ :
/$ ∇′Ei(r  v*t,t) = 1/c*∂[∂(t): Bi(r  v*t,t)]
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Question
04_27rev1
Faradays_law_differential_form
/$ ∇´Ei(r  v*t,t) = 1/c*∂[∂(t): Bi(r  v*t,t)]
/% ∇′EIodv(POIo,t) = 1/c*∂[∂(t): BIodv(POIo,t)]
/* OK  Note that this is for v=0 which seems inconsistent with "(r  v*t,t)", and I still unsure that Ive properly defined ∇′
Wasnt Stokes theorm already used in Faraday defn for (42)(418)?
If so, might this be a circular argument somehow?
(?nothing wrong with that  as it would at least "close the loop" and show consistency).
29May2016  This is OK => 14Sep2015 p69h0.95 Lucas comment that ET = E0+Ei is obscure to me at present.
13Sep2015 My approach with 1.c was NOT successful (I need to look at it again later)
/*_endCmd
/**********************************************************
/*> (428a) Faradays_law_spherical_coords  ∇´Ei(ro  vo*t,t) term
/*/*$ cat >>"$p_augmented" "$d_augment""04_28a work.txt"
25Sep2015 revision 1 started (trivial changes)
Lucas
Faradays_law_spherical_coordinates
/$ ∇´Ei(r´,t)
= r´/r/sin(θ) *[ ∂[∂O: sin(θ)*Ei(r´,t)•φ´]  ∂[∂P: Ei(r´,t)•O]
+ O/r *[1/sin(θ)*∂[∂P: Ei(r´,t)•r´]  ∂[∂r: Ei(r´,t)•φ´]
+ φ´/r *[ ∂[∂r: r*Ei(r´,t)•θ´]  ∂[∂O: Ei(r´,t)•r´]
/$ 1/c*∂[∂(t): B(r´,t)]
= v/c*r*sin(θ)*φ´*[ 3*q*v/c*(r*cos(θ)  v*t)/r  v*t^5
+ 1/r/c*∂[∂(t): Ei(r  v*t,t)] ]
/* Looking first at (427)
/$ ∇´Ei(r  v*t,t) = 1/c*∂[∂(t): Bi(r  v*t,t)]
/* I will break (428) into parts (a)&(b), which are equal to one another.
This section (428a) develops ∇´Ei(ro  vo*t,t).
/$ 1. ∇´Ei(r  v*t,t) term :
/* Generically expressing ∇´Ei(ro  vo*t,t) = ∇´Ei(r´,t) in spherical coordinates,
using Jackson1999 inside back cover (I removed common "r" to
outside of [] for e2) :
/$ ∇A
= e1*/r/sin(θ)*[ ∂[∂(θ): sin(θ)*A3)  ∂[∂(φ): A2) ]
+ e2/r *[ ∂[∂(φ): A1)/sin(θ)  ∂[∂(r): r*A3) ]
+ e3/r *[ ∂[∂(r): r*A2)  ∂[∂(θ): A1) ]
/* Use
/$ A1 = Ei(r´,t) in r´ direction = Ei(r´,t)•r´h
A2 = Ei(r´,t) in θ´ direction = Ei(r´,t)•θ´hat
A3 = Ei(r´,t) in φ´ direction = Ei(r´,t)•φ´hat
/* where the basis unit vectors are :
/$ [e1,e2,e3] = [r´h,θ´hat,φ´hat]
/* Notice that Lucase does NOT use "hat" notation for the angles
as I have used [Oh´,Ph´]  he simplifies to [Op,Pp], but to
minimize my own confusion I retain the full notation.
Also, Lucas retains the 1/r term within the partial derivatives for e2=Oh´,
and I will replace r from Jackson with rs.
/$a) ∇´Ei(r  v*t,t)
= + r´h/rs/sin(θ) *[ ∂[∂(θ): Ei(r´,t)•φ´hat*sin(θ))  ∂[∂(φ): Ei(r´,t)•θ´hat) ]
+ θ´hat/rs *[ ∂[∂(φ): Ei(r´,t)•r´h)/sin(θ)  ∂(drs: Ei(r´,t)•φ´hat*rs) ]
+ φ´hat/rs *[ ∂(drs: Ei(r´,t)•θ´hat*rs)  ∂[∂(θ): Ei(r´,t)•r´h) ]
= + r´h/rs/sin(θ) *[ ∂[∂(θ): Ei(r´,t)•φ´hat*sin(θ))  ∂[∂(φ): Ei(r´,t)•θ´hat) ]
+ θ´hat/rs *[ ∂[∂(φ): Ei(r´,t)•r´h)/sin(θ)  ∂(drs: Ei(r´,t)•φ´hat*rs) ]
+ φ´hat/rs *[ ∂(drs: Ei(r´,t)•θ´hat*rs)  ∂[∂(θ): Ei(r´,t)•r´h) ]
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Lucas_Typo_or_omission
04_28arev1
Faradays_law_spherical_coordinates  full form, 1st expression
/$ ∇´Ei(r´,t)
= + r´h/rs/sin(θ)*[ ∂[∂(θ): Ei(r´,t)•φ´hat*sin(θ))  ∂[∂(φ): Ei(r´,t)•θ´hat) ]
+ θ´hat/rs *[ ∂[∂(φ): Ei(r´,t)•r´h)/sin(θ)  ∂(drs: Ei(r´,t)•φ´hat*rs) ]
+ φ´hat/rs *[ ∂(drs: Ei(r´,t)•θ´hat*rs)  ∂[∂(θ): Ei(r´,t)•r´h) ]
/% ∇´EIodv(POIo,t) = 1/c*∂[∂(t): BIodv(POIp(t),t)]
= + Rpch(POIo(t),t) /Rocs(POIo)/sin(Aθoc(POIo)) *{ ∂[dAOoc: EIodv(POIo,t)•ROpdh*sin(Aθoc(POIo))]  ∂[dAPoc: EIodv(POIo,t)•ROpdh] }
+ ROpdh/Rocs(POIo) *{ ∂[dAPch: EIodv(POIo,t)•Rpch(POIo(t),t)/sin(Aθoc(POIo))]  ∂[dRocs(POIo): EIodv(POIo,t)*Rocs(POIo)•RPods] }
+ Rφpdh/Rocs(POIo) *{ ∂[dRocs(POIo): EIodv(POIo,t)•ROpdh•ROpdh]  ∂[dAOod: EIodv(POIo,t)•Rpch(POIo(t),t)] }
/* OK  straightforward, with a couple of concerns with Lucas's expression  reference frames & notational.
Note that I have taken 1/r out of the [] for Oh´
plus I retain [Oh´,Ph´] within []  Lucas simplifies to [Op,Pp]
But  shouldnt all angles be primed to get particle/system refFrame?
shouldnt there be a hat.over.last AOpdh in the second term?
/*_endCmd
/**********************************************************
/*> (428b) Faradays_law_spherical_coords  1/c*dp[dt : Bi(ro  vo*t,t)] term
25Sep2015 revision 1 started (trivial changes)
1. Use 04_16 E,B for symmetry point charge @v_const
Derivative form of Amperes law
/$L 1/c*∂[∂(t): Bi(r  v*t,t)]
= vs/c*rs*sin(θ)*φ´*[3*q*vs/c*(rs*cos(θ)  vs*t)/rs  vs*t^5 + 1/rs/c*∂[∂(t): Eis(r  v*t,t)] ]
/$H 1/c*∂[∂(t): Bi(r  v*t,t)]
= vs/c*rs*sin(θ)*φ´*[3*q*vs/c*(rs*cos(θ)  vs*t)/rs  vs*t^5 + 1/rs/c*∂[∂(t): Eis(r  v*t,t)] ]
/* Simple! used 416, can insert minus signs to fit Lucas format
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
04_28brev1
Faradays_law_spherical_coordinates  full form
/$L ∇´Ei(r  v*t,t) = 1/c*∂[∂(t): B(r´,t)]
= vs/c*rs*sin(θ)*φ´*[ 3*q*vs/c*(rs*cos(θ)  vs*t)/r  v*t^5 + 1/rs/c*∂[∂(t): Eis(r  v*t,t)] ]
/$H ∇´Ei(r  v*t,t) = 1/c*∂[∂(t): B(r´,t)]
= ??? vs/c*rs*sin(θ)*φ´*[ 3*q*vs/c*(rs*cos(θ)  vs*t)/r  v*t^5 + 1/rs/c*∂[∂(t): Eis(r  v*t,t)] ]
/% ∇´Ei(ro  vo*t,t) = 1/c*∂[∂(t): BIodv(POIo,t)]
= Vons(PART)/c*Rocs(POIo)*sin(Aθoc(POIo))*RAPpdh
*{ 3*Q(particle)*Vons(PART)/c*(Rocs(POIo)*cos(Aθoc(POIo))  Vons(PART)*t)/Rpcs(POIo(t),t)^5
+ 1/Rpcs(POIo(t),t)/c*∂[∂(t): EIods(POIo,t)]
}
/* OK  Simple! I used 416. 29May2016  still a concern => But  shouldnt all angles be primed to get particle/system refFrame?
/**********************************************************
/*> (429a) Faradays_law_spherical_coords  1st term
/*/*$ cat >>"$p_augmented" "$d_augment""04_29a work.txt"
25Sep2015 revision 1 started
14Sep2015 start verification, 23Sep2015 corrections & cleanup, 27May2016 reworked
From 04_28a Faradays_law_spherical_coordinates  full form, 1st expression
/$ ∇´Ei(r´,t)
= + r´h/rs/sin(θ)*[ ∂[∂(θ): Ei(r´,t)•φ´hat*sin(θ))  ∂[∂(φ): Ei(r´,t)•θ´hat) ]
+ θ´hat/rs *[ ∂[∂(φ): Ei(r´,t)•r´h)/sin(θ)  ∂[∂(r): Ei(r´,t)•φ´hat*rs) ]
+ φ´hat/rs *[ ∂[∂(r): Ei(r´,t)•θ´hat*r)  ∂[∂(θ): Ei(r´,t)•r´h) ]
/* By symmetry, the following partial derivatives are zero (with respect to the scalar result!!):
/$ ∂[∂(φ): Ei(r´,t)•θ´hat)
∂[∂(φ): Ei(r´,t)•r´h)
∂[∂(θ): Ei(r´,t)•r´h)
/* The following are zero because Ei(r´,t) & Ph are at right angles :
/$ ∂[∂(θ): Ei(r´,t)•φ´hat*sin(θ))
∂[∂(r): Ei(r´,t)•φ´hat*rs)
/* Therefore, the ONLY partial derivative that remains is :
/$ ∂[∂(θ): Ei(r´,t)•r´h)
/* yielding the expression :
/$ ∇´Ei(r´,t) = φ´hat/rs*∂[∂(θ): Ei(r´,t)•r´h)
/* but rh is a unit vector in the same direction as Ei, so for a partial derivative along angle O :
/$ ∂[∂(θ): Ei(r´,t)•r´h) = ∂[∂(θ): Ei(r´,t))
/*so
/$ ∇´Ei(r´,t) = φ´hat/rs*∂[∂(θ): Ei(r´,t))
/* which is the same as Lucas's LHS expression.
/* Earlier approach, essentially the same, but clouded by ambiguous wording in Lucas's explanation
Extract the "dp[dO : " terms :
/$ ∂[∂(θ): Ei(r´,t))
= + r´h/rs/sin(θ)*[ ∂[∂(θ): Ei(r´,t)•φ´hat*sin(θ)) ]  φ´hat/rs *[ ∂[∂(θ): Ei(r´,t)•r´h) ]
= + r´h/rs/sin(θ)*[ ∂[∂(θ): Ei(r´,t)•φ´hat*sin(θ)) ]  φ´hat/rs *[ ∂[∂(θ): Ei(r´,t)•r´h) ]
/*zz)see "Howell  Variables, notations, styles for Bill Lucas, Universal Force.odt" :
zz)"Pseudoconstant variables/expressions" within integrals wrt O
zz)Remove : [q,v,L(v),r´,c,b]
zz)Retain : [sinO,O] are the key retained variables for Chapter 4 verifications
/$1) ∂[∂(θ): Ei(r´,t))
= + r´h/rs/sin(θ)•φ´hat*[ ∂[∂(θ): Ei(r´,t)*sin(θ)) ]
 φ´hat/rs *[ ∂[∂(θ): Ei(r´,t)•r´h) ]
/* Now
/$ ∂[∂(θ): Ei(r´,t)*sin(θ)) = ∂[∂(θ): Ei(r´,t))*sin(θ) + Ei(r´,t)*∂[∂(θ): sin(θ))
∂[∂(θ): Ei(r´,t)*sin(θ)) = ∂[∂(θ): Ei(r´,t))*sin(θ) + Ei(r´,t)*cos(θ)
/* But from symmetry
/$ ∂[∂(θ): Ei(r´,t)*sin(θ)) = ∂[∂(θ): Ei(r´,t))*sin(θ) + Ei(r´,t)*cos(θ)
∂[∂(θ): Ei(r´,t)) = 0
/* therefore
/$1a) ∂[∂(θ): Ei(r´,t)*sin(θ)) = Ei(r´,t)*cos(θ)
1b) ∂[∂(θ): Ei(r´,t)•r´h) = 0 given symmetry
/* Sub (1a),(1b) into (1)
/$1) ∂[∂(θ): Ei(r´,t))
= + r´h/rs/sin(θ)•φ´hat*[ ∂[∂(θ): Ei(r´,t)*sin(θ)) ]  φ´hat/rs*[ ∂[∂(θ): Ei(r´,t)•r´h) ]
= + r´h/rs/sin(θ)•φ´hat*Ei(r´,t)*cos(θ)  φ´hat/rs*0
= + r´h/rs/sin(θ)•φ´hat*Ei(r´,t)*cos(θ)
/* The RHS of (429a) is the exact same as (428b) and (416) so no
further work is required on that part.
/$ 1/c*∂[∂(t): Bi(r  v*t,t)]
= vs/c*rs*sin(θ)*φ´*[ 3*q*vs/c*(rs*cos(θ)  vs*t)/rs  vs*t^5 + 1/rs/c*∂[∂(t): Eis(r  v*t,t)] ]
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
04_29arev1
Faradays_law_spherical_coordinates  reduced, integral form
/$L 1/rs*∂[∂(θ): Ei(r  v*t,t))*φ´ = vs/c*rs*sin(θ)*φ´*[ 3*q*vs/c*(r*cos(θ)  v*t)/r  v*t^5 + 1/rs/c*∂[∂(t): Eis(r  v*t,t)] ]
/$H 1/rs*∂[∂(θ): Ei(r  v*t,t))*φ´ = vs/c*rs*sin(θ)*φ´*[ 3*q*vs/c*(r*cos(θ)  v*t)/r  v*t^5 + 1/rs/c*∂[∂(t): Eis(r  v*t,t)] ]
/* OK  straightforward, but clouded by some of the text explanations & omissions.
/% 1/Rocs(POIo)*∂[∂(θ): EIods(POIo,t)]*Rφpdh
= Vons(PART)/c*Rocs(POIo)*sin(Aθoc(POIo)) *Rφpdh
*{ 3*Q(particle)*Vons(PART)/c*(Rocs(POIo)*cos(Aθoc(POIo))  Vons(PART)*t)/Rpcs(POIo(t),t)^5
+ 1/Rocs(POIo)/c*∂[∂(t): EIods(POIo,t)]
}
/*_endCmd
/**********************************************************
/*> (429b) Faradays_law_spherical_coords  2nd term
25Sep2015 revision 1 started, 27May2016 added quick explanation
From the verification of (429a) above :
By symmetry, the following partial derivatives are zero (with respect to the scalar result!!):
/$ ∂[∂(φ): Ei(r´,t)•θ´hat)
∂[∂(φ): Ei(r´,t)•r´h)
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
04_29brev1
Faradays_law_spherical_coordinates  dropping term
/$ 1/r/sin(θ)*∂[∂Pp: Ei(r  v*t,t)*θ´hat] = 0
1/r/sin(θ)*∂[∂Pp: Ei(r  v*t,t)*θ´hat] = 0
/* OK  easy after fixing (429a) above
/% 1/Rocs(POIo)/sin(Aθoc(POIo))*∂[∂(φ´): EIods(POIo,t)*Rθpch] = 0
/**********************************************************
/*> (430) Faradays_law integrated over θ
24Sep2015  With this revision, I will take extra care with variable notations (hat, prime, scalar, etc).
25Sep2015 revamped!!
Lucas's result looks wrong?
Faradays_law_integrated
/*/*$ cat >>"$p_augmented" "$d_augment""04_30 work.txt"
/$ ∫[dEi(r  v*t,t),0 to θ´: 1)
= Ei(r  v*t,t) for θ´ from 0 to θ´f
= Ei(r  v*t,t)  Ei(r  v*t,t,θ´ = 0)
1/c*∂[∂(t): Bi(r  v*t,t)]
= 3*(v/c*r)^2*q *∫[∂(θ´),0 to θ´: (r*cos(θ´)  v*t)/r  v*t^5*sin(θ´)
+ v*r^2/c *∫[∂(θ´),0 to θ´: 1/r/c*∂[∂(t): Ei(r  v*t,t)*sin(θ´))???brackets!!!!
/* therefore
/$ Ei(r  v*t,t)  Ei(r  v*t,t,θ´ = 0)
= 3*(v/c*r)^2*q *∫[∂(θ´),0 to θ´: (r*cos(θ´)  v*t)/r  v*t^5*sin(θ´)
+ v*r^2/c *∫[∂(θ´),0 to θ´: 1/r/c*∂[∂(t): Ei(r  v*t,t)*sin(θ´))???brackets!!!!
/* 1. Take Lucas's result for (429a), remove the minus signs
04_29arev1 Faradays_law_spherical_coordinates  reduced, integral form
/$1) 1/rs*∂[∂(θ´): Ei(r  v*t,t))*φ´
= vs/c*rs*sin(θ´)*φ´*[ 3*q*vs/c*(r*cos(θ´)  v*t)/r  v*t^5
+ 1/rs/c*∂[∂(t): Eis(r  v*t,t)] ]
/* Taking (1), expand while keeping a "*1/rs/c" term with "dp[dt : Eis]", drop  signs
/$ 1/rs*∂[∂(θ´): Ei(r  v*t,t))*φ´
= + vs/c *rs*sin(θ´)*φ´* 3*q*vs/c*(rs*cos(θ´)  vs*t)/r  v*t^5
+ vs/c *rs*sin(θ´)*φ´* 1/rs/c*∂[∂(t): Eis(r  v*t,t)]
= + 3*(vs/c)^2*rs*sin(θ´)*φ´*q *(rs*cos(θ´)  vs*t)/r  v*t^5
+ vs/c *rs*sin(θ´)*φ´ *1/rs/c*∂[∂(t): Eis(r  v*t,t)]
/* Integrate this, switching again from Op to Op? (MUST check later!)
/$ ∫[∂(θ´),0 to θ´f:  1/rs*∂[∂(θ´): Ei(r  v*t,t))*φ´]
= ∫[∂(θ´),0 to θ´f: 3*(vs/c)^2*rs*sin(θ´)*φ´*q*(rs*cos(θ´)  vs*t)/r  v*t^5]
∫[∂(θ´),0 to θ´f: vs/c*rs*sin(θ´)*φ´*1/rs/c*∂[∂(t): Eis(r  v*t,t)]]
/*zz)see "Howell  Variables, notations, styles for Bill Lucas, Universal Force.odt" :
zz)"Pseudoconstant variables/expressions" within integrals wrt Op
/$zz)Remove : [q,v,λ(v),r´,c,β]
zz)Retain : [sin(θ´),θ´] are the key retained variables for Chapter 4 verifications
1/rs*φ´ *∫[∂(θ´),0 to θ´f: ∂[∂(θ´): Ei(r  v*t,t))]
= 3*(vs/c)^2*rs*φ´*q/r  v*t^5*∫[∂(θ´),0 to θ´f: sin(θ´)*(rs*cos(θ´)  vs*t)]
+ vs/c *rs*φ´ *∫[∂(θ´),0 to θ´f: sin(θ´)/rs/c*∂[∂(t): Eis(r  v*t,t)]]
/* HOWEVER, Lucas prefers to leave ro  vo*t^5 within the integral, so
Ill reinsert it :
/$ 1/rs*φ´ *∫[∂(θ´),0 to θ´f: ∂[∂(θ´): Ei(r  v*t,t))]
= 3*(vs/c)^2*rs*φ´*q*∫[∂(θ´),0 to θ´f: (rs*cos(θ´)  vs*t)*sin(θ´)/r  v*t^5]
+ vs/c *rs*φ´ *∫[∂(θ´),0 to θ´f: 1/rs/c*∂[∂(t): Eis(r  v*t,t)]*sin(θ´)]
/* Cancel variables LHS vs RHS
/$ ∫[∂(θ´),0 to θ´f: ∂[∂(θ´): Ei(r  v*t,t))]
= 3*(vs/c*rs)^2*q *∫[∂(θ´),0 to θ´f: (rs*cos(θ´)  vs*t)*sin(θ´)/r  v*t^5]
+ vs/c*rs^2 *∫[∂(θ´),0 to θ´f: 1/rs/c*∂[∂(t): Eis(r  v*t,t)]*sin(θ´)]
/* While dp(dOp : Ei(ro  vo*t,t)) is nominally zero for the simple case of
a single, stationary point charge, it will not be in general. So taking
the integral of the derivative simply leaves me with :
/$ Ei(r  v*t,t)θ´=θ´f  Ei(r  v*t,t,θ´ = 0)
= 3*(vs/c*rs)^2*q*∫[∂(θ´),0 to θ´f: (rs*cos(θ´)  vs*t)/r  v*t^5*sin(θ´)]
+ vs/c*rs^2 *∫[∂(θ´),0 to θ´f: 1/rs/c*∂[∂(t): Eis(r  v*t,t)]*sin(θ´)]
/* but
/$ Rpcs(POIo(t),t)*sin(AOpc(POIo,t)) = Rocs(POIo)*sin(AOoc(POIo))
= 3*(vs/c*rs)^2*q*∫[∂(θ´),0 to θ´f: (r*cos(θ)  vs*t)/r  v*t^5*sin(θ´)]
+ vs/c*rs^2 *∫[∂(θ´),0 to θ´f: 1/rs/c*∂[∂(t): Eis(r  v*t,t)]*sin(θ´)]
/* Im NOT comfortable with this!!
For the overall solution approach, the Eis integral is left untouched.
Eventually, it will be argued that successive iterations result in smaller &
smaller terms that can be dropped from a series expansion.
/* 30Aug2019 see "Howell  Background math for Lucas Universal Force, Chapter 4.txt"
section '"Rpcs(POIo(t),t)" is a constant for integrals like "∫{∂(Aθtc),0 to Aθoc(POIp(t),t=0):" ?'
/% EIods(POIp(t),t)  EIods(POIp(t),t=0)
= 3*Q*Vons(PART)^2/c^2*Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))*Rpcs(POIo(t),t=0)^(  5)
*{ Rocs(POIo)*cos(Aθpc(POIo(t),t))  Vons(PART)*t }
]
+ Vons(PART) /c *Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*∂[∂(t): EIods(POIo,t)]*sin(Aθpc(POIo(t),t)) ]
/* Clearly, the integral with respect to ∂(Aθpc),0 to Aθpc(POIo(t),t=0) is taken at a "snapshot of time", so t is a constant, independent, for the purposes of this integral, from ∂(Aθpc).
The expression above becomes :
/% EIods(POIp(t),t)  EIods(POIp(t),t=0)
= 3*Q*Vons(PART)^2/c^2*Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(5)
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))
*{ Rocs(POIo)*cos(Aθpc(POIo(t),t))  Vons(PART)*t }
]
+ Vons(PART) /c *Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*∂[∂(t): EIods(POIo,t)]*sin(Aθpc(POIo(t),t)) ]
/* ???>
/* 30Aug2019 (430) Shouldn't this be a full integral [2*PI = Aθpc(POIo(t),t=0) for 2D, or spherical surface]? :
∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): Rpcs(POIo(t),t)^(6)*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t)) ]
eg. ∂(Aθpc)/∂(2*PI) > fraction of a uniform
No? This is not an integral in time, it is an integral over a spherical surface in the direction of θ at an instant in time. Aθpc(POIo(t),t) = 0 is point either [towards, backwards] the direction of motion of the particle in RFo.
Yes? These equations are really only set up for constant normal [fields, forces] at a constant radius Rpcs(POIo(t),t=0)
/* <???
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn OK  just take Rpcs(POIo(t),t=0) out of the integral as a constant wrt θ
04_30rev4
Lorentz force  Faradays_law_integrated
/$L Ei(r  v*t,t)θ´=θ´f  Ei(r  v*t,t,θ´ = 0)
= 3*(vs/c*rs)^2*q*∫[∂(θ´),0 to θ´f: (rs*cos(θ´)  vs*t)/r  v*t^5*sin(θ´)]
+ vs/c*rs ^2 *∫[∂(θ´),0 to θ´f: 1/rs/c*∂[∂(t): Eis(r  v*t,t)]*sin(θ´)]
/$H Ei(r  v*t,t)θ´=θ´f  Ei(r  v*t,t,θ´ = 0)
= 3*(vs/c*rs)^2*q*∫[∂(θ´),0 to θ´f: (rs*cos(θ´)  vs*t)/r  v*t^5*sin(θ´)]
+ vs/c*rs ^2 *∫[∂(θ´),0 to θ´f: 1/rs/c*∂[∂(t): Eis(r  v*t,t)]*sin(θ´)]
/* OK  simple, but I must redo with "current" derivations I have.
Note that the derivative in the second line below has RFo notation (important to keep me on track!).
The expression below may unnecessarily restrict the integrand to t=0 even though that is the ultimate context?
/* ???> 27Aug2019 Where Lucas uses r*cos(θ'), I am using Rocs(POIo)*cos(Aθpc(POIo(t),t))
as that is the only thing that makes sense to me at this stage! but it doesn't make sense!!
WRONG? > Lucas seems to be using θ' (prime) as an arbitrary integration variable, which is a VERY BAD idea!!??!!
<???
/%
(mathL)/* (430) /%
EIods(POIp(t),t)  EIods(POIp(t),t=0)
= 3*Q*Vons(PART)^2/c^2*Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(5)
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))
*{ Rocs(POIo)*cos(Aθpc(POIo(t),t))  Vons(PART)*t }
]
+ Vons(PART) /c *Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*∂[∂(t): EIods(POIo,t)]*sin(Aθpc(POIo(t),t)) ]
(431) From Lenzs law and symmetry of local forces
Lucas  From Lenzs law and symmetry of local forces,
/$ Eis(r  v*t,t)@O=0
/* should oppose the induced field
/$ Eis(r  v*t,t)r´/r´
/* which is proportional to the moving static field
/$ E0s(r  v*t,t)r´/r´´
/* OK with concerns  Looks reasonable (see caveats below) and straightforward. No details needed here.
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Problem_or_challenge
04_31
F therefore E balance from Lenzs law and symmetry of local forces
/$L Eis(r  v*t,t)(O=0)*r´h = λ(v)*E0s(r  v*t,t)*r´h
/$H Eis(r  v*t,t)(O=0)*r´h = λ(v)*E0s(r  v*t,t)*r´h
/* OK with concerns  Looks reasonable (see caveats below) and straightforward, but doesnt Lenzs Law apply to other angles as well? No details needed here.
Lenzs Law seems to be a very general, only referring to a proportionality between E0 and EI, but Lucas is not providing any other functional relations. Here Lucas has taken it quite literally!?
I do not need to do work here, but I am a bit uncomfortable with this simplification, which has huge implications later
/% EIods(POIo,t,Aθpc(POIp) = 0)*Rpch(POIo(t),t) = λ(Vons(PART))*E0ods(POIo,t)*Rpch(POIo(t),t)
/* Notice that the static component E0 isn't direction dependent, just distance
The above relation also implies that
/% EIods(POIo,t,Aθpc(POIp) = 0) = λ(Vons(PART))*E0ods(POIo,t)
/**********************************************************
/*> (431a) Machs principle  Lenz works, SRT & covariant Maxwell fail
Lucas makes statement that the use of Lenzs law satifies Machs principle,
whereas neither Einsteins Special Relativity Theory (SRT) nor the covariant
form of Maxwells equations do.
p70h0.
/*+++++++++++
/*add_eqn "Howell_incomplete
04_31a
Machs principle  Lenz works, SRT & covariant Maxwell fail
Provided as statement only
not done yet  Important issue for Lucas  I"ll have to think this over...
Machs principle  Important issue for Lucas. not done yet. I"ll have to think this over...
/**********************************************************
>>>>>> Summary  Derivation of the relativistic correction factor (1  β^2*sin(Aθpc(POIo(t),t=0))^2)^(3/2)
/*/*$ cat >>"$p_augmented" "$d_Lucas""context/ddt Rpcs^5*t*_cos  1 comments.txt"
In Lucas's "Universal Force" theory, the expression ∂[∂(t): Rpcs(POIo(t),t)^(5)*t*(cos(Aθpc(POIo(t),t))  1)] has an important influence over the derivation of the "relativistic correction factor" :
{1  (β*sin(Aθpc(POIo(t),t=0)^n}^(3/2)
A key objective of equations 432 through 437 is to derive that relativistic correction factor.
Lucas sets ∂[∂(t): Rpcs(POIo(t),t)^(5)*t*(cos(Aθpc(POIo(t),t))  1)] to zero, in which case :
 the correct functional forms of (β*sin(Aθpc(POIo(t),t=0)^n are obtained
 the coefficients of the series [are divergent, are not binomial series]
I personally cannot justify arbitrarily setting ∂[∂(t): Rpcs(POIo(t),t)^(5)*t*(cos(Aθpc(POIo(t),t))  1)] to zero within the context of Lucas's theories. In other words, I have not been able to get the same result as Lucas. That's not a big deal for me, as I suspect that the relativistic correction factor may be more of an [instrumentation, measurement, observation] issue rather than a basic phenomena of electromagnetism.
My attempts up to 16Oct2019 were based on applying an [iterative, nonfeedback] approach similar to Newton's method for solving implicit equations. Two variants were attempted :
1. Noninclusion of expressions with (cos(Aθpc(POIo(t),t=0))  1)
 Once I corrected numerous mistakes I had made, these led to the proper functional result, but the coefficients of the infinite series were constantly expanding.
 see subdirectory "$d_Lucas""formulae Lucas/cos  1 no, iterative, nonfeedback/"
2. Noninclusion of expressions with (cos(Aθpc(POIo(t),t=0))  1)  as recommended by Lucas
 The proper functional form was NOT obtained. Again, the coefficients of the infinite series were constantly expanding.
 I do NOT have any solid reason (certainly no proof) that allows the (cos  1) terms to be dropped! Severely restrictive (sometimes inconcistent) can lead to this, but I am not at all comfortable with these!
 see subdirectory "$d_Lucas""formulae Lucas/cos  1 yes, iterative, nonfeedback/"
In neither case above were the binomial series coefficients obtained.
16Oct2019 Status
That's it. I give up on the relativistic correction factor, from an [iterative, nonfeedback] perspective, even though it is quite likely that I have made [simple, fundamental] errors.
Only versions of 432 through 437 that drop expressions with (cos(Aθpc(POIo(t),t=0))  1) are shown below in this document (this could change in the future!). However, you can compare results by looking at :
"$d_Lucas""formulae Lucas/cos  1 no, iterative, nonfeedback/Lucas 43237 no cos  1.txt"
"$d_Lucas""formulae Lucas/cos  1 yes, iterative, nonfeedback/Lucas 43237 with cos  1.txt"
But as per the section "Multiple conflicting hypothesis" below, other approaches may also be considered.
I currently think that "∂[∂(t): Rpcs(POIo(t),t)^(5)*t*(cos(Aθpc(POIo(t),t))  1)]" DEFINITELY should be included in derivations, but that :
 is based on a iterative solution, rather than an electromagnetic feedback effect as per Thomas Barnes (Lucas's source)
 ruins the functional form of results for the relativistic correction factor
 the coefficients of the series [are divergent, are not binomial series]
Note that [special, general] relativity does provide an explanation, but :
 in a nonphenomenological sense (more like datafitting)
 "General relativity is a turkey"  see ?link to my webpage?
The concepts of [Thomas Barnes, Oleg Jefimenko, Ed Dowdye, Rami Ahmad ElNabulsi, others] may provide a much more solid explanation for the "relativistic correction factor", but I have not verified these concepts stepbystep as of 24Oct2019.
/*_endCmd
/*
>>>>>>>>> Highly restrictive conditions for dropping the (cos  1) terms /%
/*/*$ cat >>"$p_augmented" "$d_augment""relativistic factor, restrictive conditions.txt"
relativistic factor, HIGHLY restrictive conditions!!!
 RFp = RFo, particle is at origin (both reference frames)
 t=0
 Rocs(POIo) = Rpcs(POIo(t),t=0) (careful! ...)
 This does NOT apply to :
 a particle moving towards the origin of [RFo, RFp] (only motion away from origin)
 motions perpendicular to path of particle
 rotations
 ?nyet??? or a POIo moving in the direction of flight of the particle, so
 cos(Aθpc(POIo(t),t)) = 1
 sin(Aθpc(POIo(t),t)) = 0
As K1(t=0) = 0, and ∂[∂(t): K1(Aθpc(POIo(t),t=0) = 0) then no K1related terms are passed on, it can be ignored
in [steps, iterations] subsequent to (433).
/*++
/*_endCmd
/*
>>>>>>>>> Targeted results /%
/* 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}
(mathL)/* theoretical target  binomial series
n for (β*sin)^n 0 2 4 6 8 10 12
factors :
binomial series (John Wallis) = 1 3/2 3/8 1/16 3/128 3/256 7/1024
Lucas = 1 3/2 15/8 35/16
12Oct2019 Howell  no (cos  1) term, iterative, nonfeedback, based on EIods(POIo,t,4th stage)
ETpds(POIp) = 1 3/2 21/8 35/8 455/64
ETpds(POIp)*λ(Vons(PART)) = 1 1 5/4 5/3
(endMath)
/*
>>>>>>>>> Multiple conflicting hypothesis /%
Can the Universal Force be salvaged?
POSSIBLY!! Check the following multiple conflicting hypothesis :
1. use Barnes' feedback (poorly explained by Lucas)
2. Jeffimenko's causality (time) basis
3. Ed Dowdye's Extiction Light principle
4. experimentalonly  but just for particle colliders, NOT [GPS, interferometer]
At present, I have not completed stepbystep derivations for theswe additional hypothesis
/*
>>>>>>>>> Does the relativistic correction factor matter? /%
The relativistic correction factor arose from the Lorentz trnaformations, and the LorentzPoincare original theory of relativity. Presumably, the relativistic correction factor is required by at least some experiments, although history has shown that that is NOT the case for interferometer experiments, and at least some of the GPS systems.
If we simply use the factor as an experimental requirement, ignoring the many very [different, conflicting] possible conceptual explanations, that obviously removes the problem in practice. But would that compromise the rest of the Universal Force concept?
Maybe not?
/**********************************************************
/*> (432) EIods(POIo,t=0,1st stage), F therefore E balance  iteration #1 on (430)
initial  15Sep2015, rev1  ??, rev2  25Sep2015
15Sep2015 p71h0.33 Iterate [(431) into (430)] to obtain solution.
09Jun2016rev5 use HFLN, corrected (I hope) [reference frames, notations, derivatives, integrals]
14Jun2016rev6 fix my error (cosOp  1) !!
22Aug2019 remove (cosOp  1) term
30Sep2019 corrections, cleanup
/#_file_insert "cos  1 yes, iterative, nonfeedback/04_32 work.txt"
/*/*$ cat >>"$p_augmented" "$d_augment""cos  1 $cos_inclusion, iterative, nonfeedback/04_32 work.txt"
Following dedication in Lucas's book "Universal Force : Volume 1" :
"... Thomas L. Barnes, professor of Physics at the University of
Texas at El Paso, who showed the way to eliminate Einsteins Special
Relativity Theory from electrodynamics by taking into account the electrical
feedback effects of finitesized charge particles. ..."
1. using Lucas's (430) but "Howells FlatLiner Notation" (HFLN)  Lucas instructs iterative substitutions
for Ei(ro  vo*t,t) at t=0 implicitly, and dropping v*t*(cosO  1)=0t=0 terms as we go.
Presumably, at t=0 cosθ = 1, so (cosO  1)t=0 = 0
Notice that the expression variables are ALL scalar
04_30rev4 Lorentz force  Faradays_law_integrated
/* Howells FlatLiner Notation (HFLN)
From (430)
/% EIods(POIp(t),t)  EIods(POIp(t),t=0)
= 3*Q*Vons(PART)^2/c^2*Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))*Rpcs(POIo(t),t=0)^(  5)
*{ Rocs(POIo)*cos(Aθpc(POIo(t),t))  Vons(PART)*t }
]
+ Vons(PART) /c *Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*∂[∂(t): EIods(POIo,t)]*sin(Aθpc(POIo(t),t)) ]
/* rearranging for ease of view and work :
/%1) EIods(POIp(t),t)  EIods(POIp(t),t=0)
1a) = 3*Q(PART)*Vons^2/c^2*Rocs(POIo)^2*
1a1) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0):
1a1a) { Rocs(POIo)^1*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t=0)^(5)*sin(Aθpc(POIo(t),t))
1a1b)  Vons(PART)*t *Rpcs(POIo(t),t=0)^(5)*sin(Aθpc(POIo(t),t))
1a2) }
1b) + Vons(PART) /c *Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*∂[∂(t): EIods(POIo,t)]*sin(Aθpc(POIo(t),t)) ]
/* Substitute for (1a1) through (1a2) from corresponding "Bottom part" results below
/%2) EIods(POIp(t),t)  EIods(POIp(t),t=0)
2a) = 3*Q(PART)*Vons^2/c^2*Rocs(POIo)^2*
2a1)
2a1a) (+ Rocs(POIo) *Rpcs(POIo(t),t=0)^(5)* sin(Aθpc(POIo(t),t=0))^2/2
2a1b) + Vons(PART)*t*Rpcs(POIo(t),t=0)^(5)*(cos(Aθpc(POIo(t),t=0))  1)
2a2) )
2b) + Vons(PART) /c *Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*∂[∂(t): EIods(POIo,t)]*sin(Aθpc(POIo(t),t)) ]
/* Substitute Vons/c = β, factor out *Rpcs(POIo(t),t=0)^(5)
/%3) EIods(POIp(t),t)  EIods(POIp(t),t=0)
3a) = 3*Q(PART)*β^2 *Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(5)*
3a1a) (+ 1/2 *Rocs(POIo) * sin(Aθpc(POIo(t),t=0))^2
3a1b) + Vons(PART)*t *(cos(Aθpc(POIo(t),t=0))  1)
3a1c) )
3b) + β^1 *Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*∂[∂(t): EIods(POIo,t)]*sin(Aθpc(POIo(t),t)) ]
/* From (431)
/% EIods(POIo,t,Aθpc(POIp) = 0)*Rpch(POIo(t),t) = λ(Vons(PART))*E0ods(POIo,t)*Rpch(POIo(t),t)
/* Notice that the static component E0 isn't direction dependent, just distance
The above relation also implies that
/%
4) EIods(POIo,t,Aθpc(POIp) = 0) = λ(Vons(PART))*E0ods(POIo,t)
/* This must be converted to t=0 basis to be compatible, but it is on Aθpc(POIp) basis!!???
No idea of subtleties! Just use t basis that corresponds to Aθpc(POIp) basis?! (not changes with Aθpc(POIp)!?)
/* substitute (4) into (3) above /%
5) EIods(POIo,t=0) + λ(Vons(PART))*E0ods(POIo,t=0)
5a) = 3*Q(PART)*β^2 *Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(5)
5a1a) *(+ 1/2 *Rocs(POIo) * sin(Aθpc(POIo(t),t=0))^2
5a1b) + Vons(PART)*t *(cos(Aθpc(POIo(t),t=0))  1)
5a1c) )
5b) + β*Rocs(POIo)^2*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*∂[∂(t): EIods(POIo,t)]*sin(Aθpc(POIo(t),t)) ]
/* rearranging /%
6) EIods(POIo,t=0,1st stage)
= + 3 *β^2*Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(5) *1/2*Rocs(POIo)*sin(Aθpc(POIo(t),t=0))^2
+ 3 *β^2*Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(5) *Vons(PART)*t *(cos(Aθpc(POIo(t),t=0))  1)
 λ(Vons(PART))*E0ods(POIo,t=0)
+ β*Rocs(POIo)^2*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))*∂[∂(t): EIods(POIo,t)] ]
/* Compact form : /%
7) EIods(POIo,t=0,1st stage) = K_1st + f_sphereCapSurf(EIods(POIo,t))
/*_file_insert_path "$d_Lucas""relativistic factor, intermediate symbols.txt"
/****************************************************************************
>>>>>>>>> Bottom part :
/******+
/* Looking at (1a1) through (1a2) :
/%1a1) ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0):
1a1a) Rocs(POIo)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t=0)^(5)*sin(Aθpc(POIo(t),t))
1a1b)  Vons(PART)*t *Rpcs(POIo(t),t=0)^(5)*sin(Aθpc(POIo(t),t))
1a2) ]
/* Distribute the integral
/%1a1)
1a1a) ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): Rocs(POIo)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t=0)^(5)*sin(Aθpc(POIo(t),t))}
1a1b) ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0):  Vons(PART)*t*Rpcs(POIo(t),t=0)^(5)*sin(Aθpc(POIo(t),t))}
1a2)
/* Integration with respect to "AOtc" from 0 to AOtc(RFt) along curve (not a time derivative!)
see "/media/bill/PROJECTS/Lucas  Universal Force/Howell  Background math for Lucas Universal Force, Chapter 4.txt", "Derivatives & Integrals adapted to Chapter 4"
Remove from integral : [c,β,Q(PART),lambda(v),Vonv,Rocs(POIo),Rpcs(RFt),t,E0ods(POIo,t=0) for integrals (constant during integration!)]
Retain in integral : [terms with AOtc,?E,B,??]
/* Use list of special integrals  see "Howell  Background math for Lucas Universal Force, Chapter 4.odt" for derivation
These are ONLY applicable when integrating from 0 to AOpc, otherwise the lower limit must be addressed!!!
For t=0 RFt applies (RFp & RFo coincide), a,b >=0, integers. POIo does NOT have to be on the perpendicular running through the origins.
For integrals from AOtc=0 to AOtc(POIo,t=0), at AOtc=0, sin(AOpc(RFt))=0 and the expression is 0, producing a zero lower result for definite integrals.
At AOtc(RFt) = 0 & pi/2, the expression is zero. The POIo doesnt have to be at pi/2 to the particle wrt Vonv(PART).
[1  cos(Aθpc(POIp,t=0))]=0t=0
for now assume Aθpc = Aθoc
/%1a1)
1a1a) (+ Rocs(POIo) *Rpcs(POIo(t),t=0)^(5)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)) ]
1a1b)  Vons(PART)*t*Rpcs(POIo(t),t=0)^(5)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))}
1a2) )
/* first definite integral : ∫(dOp, 0 to Opf : sinOp*cosOp ) = sin^2Op/2
/% ∫[∂(Aθpc),0 to Aθpc(POIo(t),t): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)) ]
= sin(Aθpc(POIo(t),t))^2/2 from Aθpc = 0 to Aθoc(POIp(t),t=0)
= sin(Aθpc(POIo(t),t=0))^2/2  sin(0)
= sin(Aθpc(POIo(t),t=0))^2/2
/* second definite integral : 01Oct2019  is this a NEGATIVE cos?  YES dcos = sin, dsin = cos
/% ∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))}
= (1)*cos(Aθpc(POIo(t),t)) from Aθpc = 0 to Aθoc(POIp(t),t=0)
= (1)*{ cos(Aθpc(POIo(t),t=0))  cos(0) }
= (1)*{ cos(Aθpc(POIo(t),t=0))  1 }
/* The full expression becomes /%
( + Rocs(POIo) *Rpcs(POIo(t),t=0)^(5)* sin(Aθpc(POIo(t),t=0))^2/2
 Vons(PART)*t*Rpcs(POIo(t),t=0)^(5)*(1)*{ cos(Aθpc(POIo(t),t=0))  1 }
)
/* or /%
1a1a) (+ Rocs(POIo) *Rpcs(POIo(t),t=0)^(5)* sin(Aθpc(POIo(t),t=0))^2/2
1a1b) + Vons(PART)*t*Rpcs(POIo(t),t=0)^(5)*(cos(Aθpc(POIo(t),t=0))  1)
1a2) )
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Likely_Lucas_error_or_omission
04_32
F therefore E balance  simplified (430)
/$L Eis(r  v*t,t)t=0 + λ(v)*E0s(r  v*t,t)t=0
= 3 *β*rs *q/r  v*t^5*{rs/2*sin(θ´)^2 + vs*t*(cos(θ´)  1)}t=0
+ β*rs ^2 *∫[∂(θ´),0 to θ´f: 1/rs/c*∂[∂(t): Eis(r  v*t,t)]*sin(θ´))t=0
/%L Eis(r  v*t,t)t=0 + λ(v)*E0s(r  v*t,t)t=0
= 3*(β*rs)^2*q/r  v*t^5*{rs/2*sin(θ´)^2 + vs*t*(cos(θ´)  1)}t=0
+ β*rs ^2 *∫[∂(θ´),0 to θ´f: 1/rs/c*∂[∂(t): Eis(r  v*t,t)]*sin(θ´))t=0
/* Lucas derivation  Compact form : /%
EIods(POIo,t=0) = K0 + K1 + K2 + f_sphereCapSurf(EIods(POIo,t))
/* Howell derivation  drop the K1 as it fall out and doesn't contribute at t=0, see 433 /%
(mathL) EIods(POIo,t=0,1st stage) = K_1st + f_sphereCapSurf(EIods(POIo,t))
(endMath)
/*_file_insert_path "$d_Lucas""relativistic factor, intermediate symbols.txt"
/* WRONG??
I have (β*rs)^2 rather than β*rs in first expression RHS (?), but in Lucas (433) the r^2 seems to "reappear" !?!?
# enddoc
/*_endCmd
/**********************************************************
/*> (433) EIods(POIo,t=0, 2nd stage), F therefore E balance
29May2016 I had mistakenly used Ei in place of Eis, so this has been corrected in the final result.
14Jun2016rev3 fix my error (cosOp  1) !!
Lucas's version of (432)
(432rev3 F therefore E balance  simplified (430)
22Aug2019 start revision used revamped (432), finished 26Aug2019
03Sep2019 fix error with Equation (5) below  dropped a term, K3 was incorrect then simply dropped from expressions
24Sep2019 drop K1, add "Highly restrictive conditions"
30Sep2019 correct + sign for K1
/#_file_insert "cos  1 yes, iterative, nonfeedback/04_33 work.txt"
/*/*$ cat >>"$p_augmented" "$d_augment""cos  1 $cos_inclusion, iterative, nonfeedback/04_33 work.txt"
/* USUALLY, I WORK FROM LUCAS'S RESULTS RATHER THAN MY OWN, BUT IN THIS CASE I WILL START WITH MINE
22Aug2019 Howell's version of (432) :
/%
(mathL)/* generative form /%
EIods(POIo,t,2nd stage) = K_1st + f_sphereCapSurf(EIods(POIo,t,1st stage))
(endMath)
/* 1. So what is the next step?
It is interesting to directly compare (432c), as labelled (a), with (430).
(432) does not have the same integral that was replaced in (430), so the next target appears to be the integral term in the 2nd expression on the RHS :
/%
2) ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*∂[∂(t): EIods(POIo,t)]*sin(Aθpc(POIo(t),t)) ]
/* But all that I really need to do is to take the derivative of (1) and directly put that into the integral in (1)
Taking the partial derivative of EIods(POIo,t) :
/%
3) ∂[∂(t): EIods(POIo,t,1st stage)]
= + ∂[∂(t): 3/2*β^2 *Q(PART)*Rocs(POIo)^3 *Rpcs(POIo(t),t=0)^(5) * sin(Aθpc(POIo(t),t=0))^2 ]
+ ∂[∂(t): 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): λ(Vons(PART)) *Q(PART) *Rpcs(POIo(t),t=0)^(2) ]
+ ∂[∂(t): f_sphereCapSurf(EIods(POIo,t,0th stage)) ]
/* Putting (3) into (1) Yields : /%
4) EIods(POIo,t=0)
= + K0 + K1 + K2
+ β^1*Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
*{
+ ∂[∂(t): 3/2*β^2 *Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t=0)^(5) * sin(Aθpc(POIo(t),t=0))^2 ]
+ ∂[∂(t): 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): λ(Vons(PART)) *Q(PART) *Rpcs(POIo(t),t=0)^(2) ]
+ ∂[∂(t): f_sphereCapSurf(EIods(POIo,t,0th stage)) ]
}
]
/* Lucas instructs iterative substitutions for Ei(ro  vo*t,t) at t=0 implicitly, and dropping v*t*(cosO  1) terms as we go.
But why is it still in Equation (432)??
/* using /%
t=0
∂[∂(t): Rpcs(POIo(t),t)^(5)*t*(cos(Aθpc(POIo(t),t))  1)] = 0
2590:(mathH) ∂[∂(t): Rpcs(POIo(t),t)^(5)*t*(cos(Aθpc(POIo(t),t))  1)] = 0
2304:(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)
/*_file_insert_path "$d_Lucas""relativistic factor, intermediate symbols.txt"
/* Also, as the f_sphereCapSurf(EIods(POIo,t)) term is dropped after integration, it is convenient to show it separately
This also makes the meaning of the
/%
5) EIods(POIo,t=0,2nd stage)
= + K_1st
+ β^1*Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
*{
+ ∂[∂(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) ]
}
]
+ f_sphereCapSurf{∂[∂(t): f_sphereCapSurf(EIods(POIo,t,0th stage))]}
/*_file_insert_path "$d_Lucas""relativistic factor, intermediate symbols.txt"
/* ???> (433) I DON'T GET THIS! : E0ods(POIo,t=0) = Q(PART)*Rocs(POIo)/Rpcs(POIo(t),t=0)^3
It appears that, from (48) Bi(r,v,t), Lucas has replaced E0(ro  vo*t,t) in the last term with
The Grassman form of the generalized Ampere force law is based on derivations in Appendix A (eq (A19).
(408) is the derivation of (401) from the Grassman/BiotSavart form of Amperes Law
This is derived in Appendix A...
/$ q/c*(vr´)/rs'^3 = (v/c)E0(r',t')
/* reference : Jackson 1999 p?? Eqn ?? (I lost the reference location, cant find!!
such that (in Gaussian coordinates?)
This does NOT follow! :
/$ E0(r,t) = q*r´/r´s^3 = q*r´/r  v*t^3
/* BUT  in (433), Lucas has r rather than r' in numerator, WHICH SEEMS WRONG :
/$ E0(r,t) = q*r /r´s^3 = q*r /r  v*t^3
/* translate reference frame :
/% E0ods(POIo,t=0)
= Q(PART)*Rocs(POIo)/Rpcs(POIo(t),t=0)^3
/* <???
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn question with respect to E0ods(POIo,t=0) expression!
04_33 22Aug2019 start revision, 27Aug2019 finished revision
F therefore E balance  iterations on (432)
/$L Eis(r  v*t,t) APPLY t=0 TO EACH TERM
= K0 + K2
+ β*rs^2*∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t): K0 + K2])
+ β*rs^2*∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t): K3])
/* 03Sep2019 This is old!
/$H Eis(r  v*t,t) APPLY t=0 TO EACH TERM
= K0 + K2
+ β*rs^2*∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t): K0 + K2])
+ β*rs^2*∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t): K3])
/* OK  works great by using a blend of Lucas & Howell expressions for (432). This assumes a Lucas typo in 430, dropping a power of r
EXPLAIN :
Lucas states p71h0.25 that the v*t*(cosO  1) are dropped, Presumably, at t=0 cosθ = 1, so (cosO  1)t=0 = 0.
/* Result 14Sep2019  Compact form /%
EIods(POIo,t=0,2nd stage)
= + K_1st
+ β^1*Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
*{
+ ∂[∂(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) ]
}
]
+ f_sphereCapSurf{∂[∂(t): f_sphereCapSurf(EIods(POIo,t,0th stage))]}
/*_file_insert_path "$d_Lucas""relativistic factor, intermediate symbols.txt"
/*_endCmd
/**********************************************************
/*> (434) EIods(POIo,t=0, 2nd stage), K_2nd from taking partial derivatives wrt time
for previous versions, see "Howell  Old math of Lucas Universal Force.ndf"
16Sep2015 1st or 2nd?, 20Sep2015 rev2, 25Sep2015 rev4
29May2016rev5
14May2016rev6
13Aug2019 !!! ∂[∂(x): sin(x)] = cos(x) !!! ????????????????????????? /%
27Aug2019 revamp, hopefully fixing wrong [+,] issues for (437)
There is an ambiguity in my expressions with [t, t=0]  sometimes I've left in the wrong form
03Sep2019 fix error in K2 term from 04_33 (actually  this was not a problem?)
03Sep2019 Note that I corrected the results there, which had Aθpc(POIp(t),t), which is an illegal symbol!!!
this was corrected to Aθoc(POIp(t),t) or Aθpc(POIo(t),t)
03Sep2019 no net change to end result of 04_34
24Sep2019 following changes to 433, drop K1, add "Highly restrictive conditions"
02Oct2019 fix ∂[∂(t): Rpcs(POIo(t),t)^(b)*sin(Aθpc(POIo(t),t))^a], add K_1st term
14Oct2019 fix generative form for 434 (forgot to do that earlier)
/#_file_insert "cos  1 yes, iterative, nonfeedback/04_34 work.txt"
/*/*$ cat >>"$p_augmented" "$d_augment""cos  1 $cos_inclusion, iterative, nonfeedback/04_34 work.txt"
/* Restating iterative solution :
(mathL)/* generative form /%
EIods(POIo,t,2nd stage) = K_1st + f_sphereCapSurf(EIods(POIo,t,1st stage))
(endMath)
/* using /%
1) EIods(POIo,t=0,2nd stage)
= + K_1st
+ β^1*Rocs(POIo)^2
1b) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
*{
+ ∂[∂(t): K_1st]
+ ∂[∂(t): f_sphereCapSurf(EIods(POIo,t)) ]
}
]
+ f_sphereCapSurf{∂[∂(t): f_sphereCapSurf(EIods(POIo,t,0th stage))]}
/*_file_insert_path "$d_Lucas""relativistic factor, intermediate symbols.txt"
/* Substitute for (1b) from "Bottomup (6b)" results below
EIods(POIo,t=0,2nd stage)
= + K_1st
+ 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)) ]
}
*{ + 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)) ]
}
+ f_sphereCapSurf{∂[∂(t): f_sphereCapSurf(EIods(POIo,t,0th stage))]}
7) EIods(POIo,t=0,2nd stage)
= + K_1st
+ 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)) ]
}
+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}
/***********************************
>>>>>> Bottomup (2b1)
/* Looking at (1b) /%
2b) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
*∂[∂(t):
+ 3/2*β^2 *Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t)^(5)*sin(Aθpc(POIo(t),t))^2
 λ(Vons(PART))*Q(PART) /Rpcs(POIo(t),t)^2
]
]
/* Distribute the derivative /%
3b) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo) *sin(Aθpc(POIo(t),t))
*{ + ∂[∂(t): 3/2*β^2 *Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t)^(5)*sin(Aθpc(POIo(t),t))^2 ]
 ∂[∂(t): λ(Vons(PART)) *Q(PART) /Rpcs(POIo(t),t)^2 ]
}
]
/* "Percolate" constant terms up through [derivatives, integrals] in Chapter 4 :
see "Howell  Background math for Lucas Universal Force, Chapter 4.txt"
section "Constants of [derivative, integration] expressions"
section "Rpcs(POIo(t),t)" is a constant for integrals like "∫{∂(Aθtc),0 to Aθoc(POIp(t),t=0):" ?"
[c, β, λ(Vons(PART)), Q(PART), Rocs(POIo), Vons(PART)]
[Rpcs(POIo(t),t)]
 is NOT a constant wrt ∂[∂(t):
 IS a constant wrt ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0):
/%
4b) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo) *sin(Aθpc(POIo(t),t))
*{ + 3/2*β^2 *Q(PART)*Rocs(POIo)^3*∂[∂(t): Rpcs(POIo(t),t)^(5) *sin(Aθpc(POIo(t),t))^2 ]
 λ(Vons(PART))*Q(PART) *∂[∂(t): Rpcs(POIo(t),t)^(2) ]
}
]
/* using /%
2059:(mathL) ∂[∂(t): Rpcs(POIo(t),t)^(2)] = 2*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(3)
2069:(mathL) ∂[∂(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)
/* substitute into (4b) /%
5b) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo) *sin(Aθpc(POIo(t),t))
*{ + 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)
}
]
/* [collect, rearrange] terms
/% *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
*{ + 3/2*7*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)
}
]
/* distribute the integral
/* "Percolate" constant terms up through [derivatives, integrals] in Chapter 4 :
see "Howell  Background math for Lucas Universal Force, Chapter 4.txt"
section "Constants of [derivative, integration] expressions"
section "Rpcs(POIo(t),t)" is a constant for integrals like "∫{∂(Aθtc),0 to Aθoc(POIp(t),t=0):" ?"
[c, β, λ(Vons(PART)), Q(PART), Rocs(POIo), Vons(PART)]
[Rpcs(POIo(t),t)]
 is NOT a constant wrt ∂[∂(t):
 IS a constant wrt ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0):
/%
*{ + 21/2*β^2*Rocs(POIo)^3 *Q(PART)*Vons(PART) *1/c/Rocs(POIo)*Rpcs(POIo(t),t)^(6) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^2 ]
 2 *λ(Vons(PART)) *Q(PART)*Vons(PART) *1/c/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)) ]
}
*{ + 21/2*β^2*Rocs(POIo)^2 *Q(PART)*Vons(PART) *1/c *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)) *Q(PART)*Vons(PART) *1/c/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)) ]
}
/* set Vons(PART)/c = β, collect terms /%
6b) *{ + 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)) ]
}
/* insert this result in "Topdown" section above
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Likely_Lucas_error_or_omission
04_34
F therefore E balance  taking partial derivatives wrt time
/$L Eis(r  v*t,t) APPLY t=0 TO EACH TERM
=+ 3/2*β^2*q*rs^3/r  v*t^5*sin(θ´)^2  λ(v)*q*rs/r  v*t^3
+ β *rs^2*∫[∂(θ´),0 to θ´f: sin(θ´)*
(+ 15/2*β^3*q*rs^4/r  v*t^7*sin(θ´)^2*cos(θ´)
 3 *β *q*rs^2/r  v*t^5*λ(v) *cos(θ´)
))
+ β^2 *rs^4*∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t):
∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t): Eis(r  v*t,t)])])
/$H Eis(r  v*t,t) APPLY t=0 TO EACH TERM
=+ 3/2*β^2*q*rs^3/r  v*t^5*sin(θ´)^2  λ(v)*q*rs/r  v*t^3
+ β *rs^2*∫[∂(θ´),0 to θ´f: sin(θ´)*
(+ 15/2*β^3*q*rs^3/r  v*t^7*sin(θ´)^2*cos(θ´)
 3 *β *q*rs^1/r  v*t^5*λ(v) *cos(θ´)
))
+ β^2 *rs^4*∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t):
∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t): Eis(r  v*t,t)])])
/%H
EIods(POIo,t,2nd stage)
= + K_1st
+ 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): 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): sin(Aθpc(POIo(t),t)) *cos(Aθpc(POIo(t),t)) ]
}
+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}
/*_file_insert_path "$d_Lucas""relativistic factor, intermediate symbols.txt"
/*_endCmd
/**********************************************************
/*> (435) E0ods(POIo,t) truncated expression with ONLY E0ods(POIo,t) terms
16Sep2015 1st version, ?17Sep2015? rev1, 21Sep2015 rev2, 26Sep2015 rev3
for previous versions, see "Howell  Old math of Lucas Universal Force.ndf"
??? 04Jan2016 Check if should be redone with "Howell  Key math info & derivations for Lucas Universal Force.odt"
29Aug2019 rederive based on recent work from (432) to (434)
03Sep2019 fix ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))*cos(Aθpc(POIp(t),t)) ] :
I had sin(Aθpc(POIo(t),t=0))^1/1, should have been sin(Aθpc(POIo(t),t=0))^2/2
24Sep2019 following changes to 433, drop K1, add "Highly restrictive conditions"
02Oct2019 percolate fix to ∂[∂(t): Rpcs(POIo(t),t)^(b)*sin(Aθpc(POIo(t),t))^a], note K_2nd term
/#_file_insert "cos  1 yes, iterative, nonfeedback/04_35 work.txt"
/*/*$ cat >>"$p_augmented" "$d_augment""cos  1 $cos_inclusion, iterative, nonfeedback/04_35 work.txt"
/* Objective  expression for : /%
1) ETods(POIo,t) = E0ods(POIo,t) + EIods(POIo,t)
/* Gaussian coordinates  see also 04_33 Equation (7), see [explanation, worry] below /%
2) E0ods(POIo,t) = Q(PART)*Rpcs(POIp)^(2)
/* EIods(POIo,t,2nd stage)  starting with 04_34 differentiable result /%
3) EIods(POIo,t,2nd stage)
= + K_1st
+ β*Rocs(POIo)^2
*{ + 21/2 *β^3*Q(PART)*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(6)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^3 *cos(Aθpc(POIo(t),t)) ]
 2 *λ(Vons(PART)) *β *Q(PART)/Rocs(POIo) *Rpcs(POIo(t),t)^(3)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t)) *cos(Aθpc(POIo(t),t)) ]
}
+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}
/*_file_insert_path "$d_Lucas""relativistic factor, intermediate symbols.txt"
/* using /%
2687:(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
2685:(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
4) EIods(POIo,t=0,2nd stage)
= + K_1st
+ 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
}
+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}
5) EIods(POIo,t=0,2nd stage)
= + K_1st
+ 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
 β*Rocs(POIo)^2 *2 *β *Q(PART)/Rocs(POIo) *Rpcs(POIo(t),t)^(3)*sin(Aθpc(POIo(t),t=0))^2/2 *λ(Vons(PART))
+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}
/* collect factors, rearrange using /%
1108:(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)
)
6) EIods(POIo,t=0,2nd stage)
= + Q(PART)
*( 3/2 *β^2*Rocs(POIo)^3 *sin(Aθpc(POIo(t),t=0))^2 *Rpcs(POIo(t),t=0)^(5)
+ 21/8 *β^4*Rocs(POIo)^4 *sin(Aθpc(POIo(t),t=0))^4 *Rpcs(POIo(t),t=0)^(6)
 λ(Vons(PART)) *1 *Rpcs(POIo(t),t=0)^(2)
 λ(Vons(PART)) *1 *β^2*Rocs(POIo) *sin(Aθpc(POIo(t),t=0))^2 *Rpcs(POIo(t),t=0)^(3)
)
+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}
/* what Lucas seems to have done here is to use : /%
1205:(mathH) E0ods(POIo,t) = E0pds(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2
(mathL)/* HIGHLY restrictive conditions : t=0, RFo(t=0) = Rfp(t=0), Rocs(POIo) = Rpcs(POIo(t),t=0) /%
Rocs(POIo) = Rpcs(POIo(t),t=0)
(endMath)
/* ???>
02Sep2019 Note that I find that the assumption {Rocs(POIo) = Rpcs(POIo(t),t=0)} is a HUGE restriction,
and means to me that the result is not at all general (but neither are standard theories).
<???
/* reexpress (6) using Rocs(POIo) = Rpcs(POIo(t),t=0) /%
7) EIods(POIo,t=0,2nd stage)
= + 3/2 *β^2*Q(PART)*Rpcs(POIo(t),t=0)^(2) *sin(Aθpc(POIo(t),t=0))^2
+ 21/8 *β^4*Q(PART)*Rpcs(POIo(t),t=0)^(2) *sin(Aθpc(POIo(t),t=0))^4
 λ(Vons(PART)) *1 *Q(PART)*Rpcs(POIo(t),t=0)^(2)
 λ(Vons(PART)) *1 *β^2*Q(PART)*Rpcs(POIo(t),t=0)^(2) *sin(Aθpc(POIo(t),t=0))^2
+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}
/* rearranging /%
8) EIods(POIo,t=0,2nd stage)
= + Q(PART)*Rpcs(POIo(t),t=0)^(2)
*( + 3/2 *β^2*sin(Aθpc(POIo(t),t=0))^2
+ 21/8 *β^4*sin(Aθpc(POIo(t),t=0))^4
 λ(Vons(PART)) *1
 λ(Vons(PART)) *1 *β^2*sin(Aθpc(POIo(t),t=0))^2
)
+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}
/* now using /%
1174:(mathH) E0pds(POIp) = Q(PART)/Rpcs(POIp)^2
/* substitute (2) into (8), and (8) into (1), drop f_sphereCapSurf expression /%
9) EIods(POIo,t=0,2nd stage)
= + E0ods(POIo,t) *3/2 *β^2*sin(Aθpc(POIo(t),t=0))^2
 E0ods(POIo,t) *λ(Vons(PART)) *1
+ E0ods(POIo,t) *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
/* reexpress (9) /%
10) 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 }
/* For total E, ETods(POIo,t) /%
11) ETods(POIo,t=0,2nd stage)
= E0ods(POIo,t) + EIods(POIo,t)
= + 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 }
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn 03Sep2019 NOT the same thing!!!
04_35
F therefore E balance  more iterations
all measures at t=0
/$L Ei(r)t=0
= + E0(rs) *( 3/2*β^2*sin(θ)^2 + 15/8*β^4*sin(θ)^4 )
 E0(rs)*λ(v)*(1 + 3/2*β^2*sin(θ)^2)
/$H Ei(rs,vs) all measures at t=0
= + E0(rs) *( 3/2*β^2*sin(θ´)^2 + 15/8*β^4*sin(θ´)^4 )
 E0(rs)*λ(v)*(1 + 3/2*β^2*sin(θ´)^2)
/* OK other than Lucas EI/ET mixup, Oo/Op limit used by Lucas p72h0.0? probably OK, check later
E0 is added to get ET in 436.
As part of the iterative approach, Lucas dropped the last expression with Ei(r).
This is in need of [explanation, clarification]  i.e. this is the firststep approximation only.
/* Both [induced, total] forms are shown below
/%H
/* remove "=0" from t to get :
(mathL)/* differentiable form EIods(POIo,t,2nd stage) /%
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))]}
(endMath)
(mathL)/* HIGHLY restrictive conditions : t=0, RFo(t=0) = Rfp(t=0), Rocs(POIo) = Rpcs(POIo(t),t=0) /%
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}
(endMath)
(mathL)/* HIGHLY restrictive conditions : t=0, RFo(t=0) = Rfp(t=0), Rocs(POIo) = Rpcs(POIo(t),t=0) /%
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}
(endMath)
/*_file_insert_path "$d_Lucas""relativistic factor, restrictive conditions.txt"
Note that Lucas (435) is actually for EIods(POIo,t=0), not ETods(POIo,t).
/* 03Sep2019 the numerical factors differ from Lucas  hopefully that will selffix with further iterations?
/*_endCmd
/**********************************************************
/*> (436) ETods(POIo,t) expression with ONLY ETods(POIo,t) terms
17Sep2015 2nd iteration
03Sep2019 Redone with HFLN  it now works!!!!!
/#_file_insert "cos  1 yes, iterative, nonfeedback/04_36 work.txt"
/*/*$ cat >>"$p_augmented" "$d_augment""cos  1 $cos_inclusion, iterative, nonfeedback/04_36 work.txt"
Note that Lucas (435) is actually for EIods(POIo,t=0), not ETods(POIo,t).
I did both in (435), so just refer to it for the derivations.
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn same issues as 04_35
04_36
Er in terms of E0(rs) and L(vs)
/$L ET(rs,vs) = E0(rs) + Ei(rs,vs) all measures at t=0
= E0(rs) *(1 + 3/2*β^2*sin(θ)^2 + 15/8*β^4*sin(θ)^4 )
 E0(rs)*L(vs)*(1 + 3/2*β^2*sin(θ)^2 )
/$H ET(rs,vs) = E0(rs) + Ei(rs,vs) all measures at t=0
= + E0(rs) *(1 + 3/2*β^2*sin(θ)^2 + 15/8*β^4*sin(θ)^4 )
 E0(rs)*L(vs)*(1 + 3/2*β^2*sin(θ)^2)
/* OK  very simple. /%
ETods(POIo,t=0,2nd stage)
= + E0ods(POIo,t) *{ 1 + 3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 + 21/8*β^4*sin(Aθpc(POIo(t),t=0))^4 }
 E0ods(POIo,t)*λ(Vons(PART)) *{ 1 + β^2*sin(Aθpc(POIo(t),t=0))^2 }
/* 03Sep2019 the numerical factors differ from Lucas  hopefully that will selffix with further iterations?
/*_file_insert_path "$d_Lucas""relativistic factor, restrictive conditions.txt"
/*_endCmd
**********************************************************
/*> (437) Er and the binomial series, leading to the relativistic correction factor
27Sep2019  did not finish  revamp in next attempt with updated previous equations 4[32 to 36]
27Sep2019 using updated previous equations 4[32 to 36]
02Oct2019 add [K_1st, K_2nd] terms
after fixing ∂[∂(t): Rpcs(POIo(t),t)^(b)*sin(Aθpc(POIo(t),t))^a]
fix ∂[∂(t): E0ods(POIo,t) *sin(Aθpc(POIo(t),t))^a]
11Oct2019 use corrected form of EIods(POIo,t,2nd stage) with "+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}"
Approach for 22Aug2019, 03Sep2019 after correcting recent errors in 04_35 :
1. Don't track Lucas too closely  he has wrong coefficients
2. Iterations pass through same operations as (432) through (436)
3. Howells "dp[dt : E0ods(POIo,t)] = dp[dt : E0pds(POIo,t)]  see proper E0odv(POIo,t) vector approach" from "Howell  Background math for Lucas Universal Force, Chapter 4.odt"
I was very lazy with symbols 0 many improper uses!!! Must clean up thre t=0 notational mess for [integrals, derivatives]
/#_file_insert "cos  1 yes, iterative, nonfeedback/04_37 work.txt"
/*/*$ cat >>"$p_augmented" "$d_augment""cos  1 $cos_inclusion, iterative, nonfeedback/04_37 work.txt"
/*
/* Stage 3 iteration
/* Restating iterative solution :
(mathL)/* generative form /%
EIods(POIo,t,3rd stage) = K_1st + f_sphereCapSurf(EIods(POIo,t,2nd stage))
(endMath)
/* using /%
3023:(mathL)/* differentiable form /%
EIods(POIo,t,2nd stage)
= + Q(PART)
*( + 3/2 *β^2*Rocs(POIo)^3 *sin(Aθpc(POIo(t),t=0))^2 *Rpcs(POIo(t),t=0)^(5)
+ 21/8 *β^4*Rocs(POIo)^4 *sin(Aθpc(POIo(t),t=0))^4 *Rpcs(POIo(t),t=0)^(6)
 λ(Vons(PART)) *1 *Rpcs(POIo(t),t=0)^(2)
 λ(Vons(PART)) *1 *β^2*Rocs(POIo) *sin(Aθpc(POIo(t),t=0))^2 *Rpcs(POIo(t),t=0)^(3)
)
+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}
(endMath)
1) EIods(POIo,t,3rd stage)
= + K_1st
+ f_sphereCapSurf
( + Q(PART)
*( 3/2 *β^2*Rocs(POIo)^3 *sin(Aθpc(POIo(t),t=0))^2 *Rpcs(POIo(t),t=0)^(5)
+ 21/8 *β^4*Rocs(POIo)^4 *sin(Aθpc(POIo(t),t=0))^4 *Rpcs(POIo(t),t=0)^(6)
 λ(Vons(PART)) *1 *Rpcs(POIo(t),t=0)^(2)
 λ(Vons(PART)) *1 *β^2*Rocs(POIo) *sin(Aθpc(POIo(t),t=0))^2 *Rpcs(POIo(t),t=0)^(3)
)
+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}
)
= + K_1st
+ Q(PART)*f_sphereCapSurf
( 3/2 *β^2*Rocs(POIo)^3 *sin(Aθpc(POIo(t),t=0))^2 *Rpcs(POIo(t),t=0)^(5)
+ 21/8 *β^4*Rocs(POIo)^4 *sin(Aθpc(POIo(t),t=0))^4 *Rpcs(POIo(t),t=0)^(6)
 λ(Vons(PART)) *1 *Rpcs(POIo(t),t=0)^(2)
 λ(Vons(PART)) *1 *β^2*Rocs(POIo) *sin(Aθpc(POIo(t),t=0))^2 *Rpcs(POIo(t),t=0)^(3)
)
+ f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}
/* using /%
1042:(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]]
3) EIods(POIo,t,3rd stage)
= + K_1st
+ Q(PART)*β*Rocs(POIo)^2*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
3a) *∂[∂(t):
( 3/2 *β^2*Rocs(POIo)^3 *sin(Aθpc(POIo(t),t=0))^2 *Rpcs(POIo(t),t=0)^(5)
+ 21/8 *β^4*Rocs(POIo)^4 *sin(Aθpc(POIo(t),t=0))^4 *Rpcs(POIo(t),t=0)^(6)
 λ(Vons(PART)) *1 *Rpcs(POIo(t),t=0)^(2)
 λ(Vons(PART)) *1 *β^2*Rocs(POIo) *sin(Aθpc(POIo(t),t=0))^2 *Rpcs(POIo(t),t=0)^(3)
)
]
+ f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}
/* ++
Looking at (3a)
"Percolate" constant terms up through [derivatives, integrals] in Chapter 4 :
see "Howell  Background math for Lucas Universal Force, Chapter 4.txt"
section "Constants of [derivative, integration] expressions"
[c, β, λ(Vons(PART)), Q(PART), Rocs(POIo), Vons(PART)]  are ALWAYS constants in Chapter 4
[Rpcs(POIo(t),t), E0ods(POIo,t)]  are NOT constants wrt ∂[∂(t):, ARE constants wrt ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0):
/%
3b) *( + 3/2 *β^2*Rocs(POIo)^3 *∂[∂(t): sin(Aθpc(POIo(t),t=0))^2 *Rpcs(POIo(t),t=0)^(5)]
+ 21/8 *β^4*Rocs(POIo)^4 *∂[∂(t): sin(Aθpc(POIo(t),t=0))^4 *Rpcs(POIo(t),t=0)^(6)]
 λ(Vons(PART)) *1 *∂[∂(t): *Rpcs(POIo(t),t=0)^(2)]
 λ(Vons(PART)) *1 *β^2*Rocs(POIo) *∂[∂(t): sin(Aθpc(POIo(t),t=0))^2 *Rpcs(POIo(t),t=0)^(3)]
)
/* using /%
2286:(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)
2300:(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)
2263:(mathH) ∂[∂(t): Rpcs(POIo(t),t)^(2)] = 2*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(3)
2282:(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)
3c) *( + 3/2 *β^2*Rocs(POIo)^3 * 7*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(6)
+ 21/8 *β^4*Rocs(POIo)^4 *10*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(7)
 λ(Vons(PART)) * 2*Vons(PART) *cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(3)
 λ(Vons(PART)) *β^2*Rocs(POIo) * 5*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(4)
)
/* multiply numbers /%
3d) *( + 21/2 *β^2*Rocs(POIo)^3 *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(6)
+ 105/4 *β^4*Rocs(POIo)^4 *Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(7)
 λ(Vons(PART)) *2 *Vons(PART) *cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(3)
 λ(Vons(PART)) *5 *β^2*Rocs(POIo) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(4)
)
/* substitute (3d) for (3a) /%
4) EIods(POIo,t,3rd stage)
= + K_1st
4a) + Q(PART)*β*Rocs(POIo)^2*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
*( + 21/2 *β^2*Rocs(POIo)^3 *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(6)
+ 105/4 *β^4*Rocs(POIo)^4 *Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(7)
 λ(Vons(PART)) *2 *Vons(PART) *cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(3)
 λ(Vons(PART)) *5 *β^2*Rocs(POIo) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(4)
)
+ f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}
/* ++
Looking at (4a)
"Percolate" constant terms up through [derivatives, integrals] in Chapter 4 :
see "Howell  Background math for Lucas Universal Force, Chapter 4.txt"
section "Constants of [derivative, integration] expressions"
[c, β, λ(Vons(PART)), Q(PART), Rocs(POIo), Vons(PART)]  are ALWAYS constants in Chapter 4
[Rpcs(POIo(t),t), E0ods(POIo,t)]  are NOT constants wrt ∂[∂(t):, ARE constants wrt ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0):
/%
4b) + Q(PART)*β*Rocs(POIo)^2*1/c/Rocs(POIo)
*( + 21/2 *β^2*Rocs(POIo)^3 *Vons(PART)*Rpcs(POIo(t),t)^(6) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))
+ 105/4 *β^4*Rocs(POIo)^4 *Vons(PART)*Rpcs(POIo(t),t)^(7) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))
 λ(Vons(PART)) *2 *Vons(PART)*Rpcs(POIo(t),t)^(3) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t)) *cos(Aθpc(POIo(t),t))
 λ(Vons(PART)) *5 *β^2*Rocs(POIo) *Vons(PART)*Rpcs(POIo(t),t)^(4) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))
)
/* collect powers of [sin(Aθpc(POIo(t),t)), Rocs(POIo)], move Rocs(POIo)into each line, extract Vons(PART) from each line /%
4c) + Q(PART)*β*Vons(PART)/c
*( + 21/2 *β^2*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(6) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^3 *cos(Aθpc(POIo(t),t))
+ 105/4 *β^4*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(7) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^5 *cos(Aθpc(POIo(t),t))
 λ(Vons(PART)) *2 *Rocs(POIo)^1 *Rpcs(POIo(t),t)^(3) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^1 *cos(Aθpc(POIo(t),t))
 λ(Vons(PART)) *5 *β^2*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(4) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^3 *cos(Aθpc(POIo(t),t))
)
/* using /%
362:(mathL) β = Vons(PART)/c
4d) + Q(PART)*β^2
*( + 21/2 *β^2*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(6) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^3 *cos(Aθpc(POIo(t),t))
+ 105/4 *β^4*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(7) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^5 *cos(Aθpc(POIo(t),t))
 λ(Vons(PART)) *2 *Rocs(POIo)^1 *Rpcs(POIo(t),t)^(3) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^1 *cos(Aθpc(POIo(t),t))
 λ(Vons(PART)) *5 *β^2*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(4) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^3 *cos(Aθpc(POIo(t),t))
)
/* collect powers of β /%
4e) + Q(PART)
*( + 21/2 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(6) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^3 *cos(Aθpc(POIo(t),t))
+ 105/4 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(7) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^5 *cos(Aθpc(POIo(t),t))
 λ(Vons(PART)) *2 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(3) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^1 *cos(Aθpc(POIo(t),t))
 λ(Vons(PART)) *5 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(4) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^3 *cos(Aθpc(POIo(t),t))
)
/* using /%
3099:(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
3103:(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
3095:(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
3099:(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
4f) + Q(PART)
*( + 21/2 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(6) *sin(Aθpc(POIo(t),t=0))^4/4
+ 105/4 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(7) *sin(Aθpc(POIo(t),t=0))^6/6
 λ(Vons(PART)) *2 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(3) *sin(Aθpc(POIo(t),t=0))^2/2
 λ(Vons(PART)) *5 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(4) *sin(Aθpc(POIo(t),t=0))^4/4
/* collect numbers /%
4g) + Q(PART)
*( + 21/8 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(6) *sin(Aθpc(POIo(t),t=0))^4
+ 35/8 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(7) *sin(Aθpc(POIo(t),t=0))^6
 λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(3) *sin(Aθpc(POIo(t),t=0))^2
 λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(4) *sin(Aθpc(POIo(t),t=0))^4
)
/* insert (4g) in place of (4a), this can be used for t=0 form /%
5) EIods(POIo,t,3rd stage)
= + K_1st
+ Q(PART)
*( + 21/8 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(6) *sin(Aθpc(POIo(t),t=0))^4
+ 35/8 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(7) *sin(Aθpc(POIo(t),t=0))^6
 λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(3) *sin(Aθpc(POIo(t),t=0))^2
 λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(4) *sin(Aθpc(POIo(t),t=0))^4
)
+ f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}
/* drop "=0" qualifier to get differentiable form /%
/* using in (5) /%
1108:(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)
)
6) 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
 λ(Vons(PART)) *Rpcs(POIo(t),t)^(2)
)
+ Q(PART)
*( + 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 *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))}}
7) 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))}}}
/* remove "=0" from t to get :
(mathL)/* differentiable form /%
EIods(POIo,t,3rd 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)
+ 35/8 *β^6*Rocs(POIo)^5 *sin(Aθpc(POIo(t),t))^6 *Rpcs(POIo(t),t)^(7)
 λ(Vons(PART)) *1 *Rpcs(POIo(t),t)^(2)
 λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *sin(Aθpc(POIo(t),t))^2 *Rpcs(POIo(t),t)^(3)
 λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *sin(Aθpc(POIo(t),t))^4 *Rpcs(POIo(t),t)^(4)
)
+ f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}
(endMath)
/* post[differentiation, integration] form /%
/* 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"
/%
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)
/* use this in (7), careful to put t=0 /%
7) EIods(POIo,t,3rd stage)
= + Q(PART)
*( + 3/2 *β^2 *Rpcs(POIo(t),t=0)^(2)*sin(Aθpc(POIo(t),t=0))^2
+ 21/8 *β^4 *Rpcs(POIo(t),t=0)^(2)*sin(Aθpc(POIo(t),t=0))^4
+ 35/8 *β^6 *Rpcs(POIo(t),t=0)^(2)*sin(Aθpc(POIo(t),t=0))^6
 λ(Vons(PART)) *1 *Rpcs(POIo(t),t=0)^(2)
 λ(Vons(PART)) *1 *β^2 *Rpcs(POIo(t),t=0)^(2)*sin(Aθpc(POIo(t),t=0))^2
 λ(Vons(PART)) *5/4 *β^4 *Rpcs(POIo(t),t=0)^(2)*sin(Aθpc(POIo(t),t=0))^4
)
+ f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}
= + Q(PART)*Rpcs(POIo(t),t=0)^(2)
*( + 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
 λ(Vons(PART)) *1
 λ(Vons(PART)) *1 *β^2 *sin(Aθpc(POIo(t),t=0))^2
 λ(Vons(PART)) *5/4 *β^4 *sin(Aθpc(POIo(t),t=0))^4
)
+ f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}
/* using /%
1174:(mathH) E0pds(POIp) = Q(PART)/Rpcs(POIp)^2
8) EIods(POIo,t=0,3rd stage)
= + E0pds(POIp) *3/2 *β^2*sin(Aθpc(POIo(t),t=0))^2
+ E0pds(POIp) *21/8 *β^4*sin(Aθpc(POIo(t),t=0))^4
+ E0pds(POIp) *35/8 *β^6*sin(Aθpc(POIo(t),t=0))^6
 E0pds(POIp)*λ(Vons(PART)) *1
 E0pds(POIp)*λ(Vons(PART)) *1 *β^2*sin(Aθpc(POIo(t),t=0))^2
 E0pds(POIp)*λ(Vons(PART)) *5/4 *β^4*sin(Aθpc(POIo(t),t=0))^4
+ f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}
(mathL)/* HIGHLY restrictive conditions : t=0, RFo(t=0) = Rfp(t=0), Rocs(POIo) = Rpcs(POIo(t),t=0) /%
EIods(POIo,t=0,3rd 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}
 E0pds(POIp)*λ(Vons(PART)) *{1 + β^2*sin(Aθpc(POIo(t),t=0))^2 + 5/4 *β^4*sin(Aθpc(POIo(t),t=0))^4}
(endMath)
/*_file_insert_path "$d_Lucas""relativistic factor, intermediate symbols.txt"
/*
/* Stage 4 iteration
/* Start with iteration 3 of 11Oct2019 /%
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))}}}
/* see "textIn  relativistic factor, intermediate symbols, restrictive conditions.txt"
Restating iterative solution :
/%
(mathL)/* generative form /%
EIods(POIo,t,4th stage) = K_1st + f_sphereCapSurf(EIods(POIo,t,3rd stage))
(endMath)
1) EIods(POIo,t,4th stage)
= + K_1st
+ f_sphereCapSurf
( + 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))}}}
)
= + K_1st
+ Q(PART)*f_sphereCapSurf
( + 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{f_sphereCapSurf(EIods(POIo,t,0th stage))}}}}
/* using /%
1042:(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]]
2) EIods(POIo,t,4th stage)
= + K_1st
+ Q(PART)*β*Rocs(POIo)^2*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
2a) *∂[∂(t):
( + 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{f_sphereCapSurf(EIods(POIo,t,0th stage))}}}}
/* ++
Looking at (3a)
"Percolate" constant terms up through [derivatives, integrals] in Chapter 4 :
see "Howell  Background math for Lucas Universal Force, Chapter 4.txt"
section "Constants of [derivative, integration] expressions"
[c, β, λ(Vons(PART)), Q(PART), Rocs(POIo), Vons(PART)]  are ALWAYS constants in Chapter 4
[Rpcs(POIo(t),t), E0ods(POIo,t)]  are NOT constants wrt ∂[∂(t):, ARE constants wrt ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0):
/%
2b)
( + 3/2 *β^2*Rocs(POIo)^3 *∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(5)]
+ 21/8 *β^4*Rocs(POIo)^4 *∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(6)]
+ 35/8 *β^6*Rocs(POIo)^5 *∂[∂(t): sin(Aθpc(POIo(t),t))^6*Rpcs(POIo(t),t)^(7)]
 λ(Vons(PART)) *1 *∂[∂(t): *Rpcs(POIo(t),t)^(2)]
 λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(3)]
 λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(4)]
)
/* using /%
2286:(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)
2300:(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)
2336:(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)
2263:(mathH) ∂[∂(t): Rpcs(POIo(t),t)^(2)] = 2*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(3)
2282:(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)
2314:(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)
2c) ( + 3/2 *β^2*Rocs(POIo)^3 * 7*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(6)
+ 21/8 *β^4*Rocs(POIo)^4 *10*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(7)
+ 35/8 *β^6*Rocs(POIo)^5 *13*Vons(PART)*sin(Aθpc(POIo(t),t))^6*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(8)
 λ(Vons(PART)) *1 * 2*Vons(PART) *cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(3)
 λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 * 5*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(4)
 λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 * 8*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(5)
)
/* multiply numbers /%
2d) ( + 21/2 *β^2*Rocs(POIo)^3 *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(6)
+ 105/4 *β^4*Rocs(POIo)^4 *Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(7)
+ 455/8 *β^6*Rocs(POIo)^5 *Vons(PART)*sin(Aθpc(POIo(t),t))^6*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(8)
 λ(Vons(PART)) *2 *Vons(PART) *cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(3)
 λ(Vons(PART)) *5 *β^2*Rocs(POIo)^1 *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(4)
 λ(Vons(PART)) *10 *β^4*Rocs(POIo)^2 *Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(5)
)
/* substitute (2d) for (2a) /%
3) EIods(POIo,t,4th stage)
= + K_1st
3a) + Q(PART)*β*Rocs(POIo)^2*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
*( + 21/2 *β^2*Rocs(POIo)^3 *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(6)
+ 105/4 *β^4*Rocs(POIo)^4 *Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(7)
+ 455/8 *β^6*Rocs(POIo)^5 *Vons(PART)*sin(Aθpc(POIo(t),t))^6*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(8)
 λ(Vons(PART)) *2 *Vons(PART) *cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(3)
 λ(Vons(PART)) *5 *β^2*Rocs(POIo)^1 *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(4)
 λ(Vons(PART)) *10 *β^4*Rocs(POIo)^2 *Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(5)
)
+ f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}}}
/* ++
Looking at (3a)
"Percolate" constant terms up through [derivatives, integrals] in Chapter 4 :
see "Howell  Background math for Lucas Universal Force, Chapter 4.txt"
section "Constants of [derivative, integration] expressions"
[c, β, λ(Vons(PART)), Q(PART), Rocs(POIo), Vons(PART)]  are ALWAYS constants in Chapter 4
[Rpcs(POIo(t),t), E0ods(POIo,t)]  are NOT constants wrt ∂[∂(t):, ARE constants wrt ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0):
/%
3b) + Q(PART)*β*Rocs(POIo)^2
*( + 21/2 *β^2*Rocs(POIo)^3 *Vons(PART)*Rpcs(POIo(t),t)^(6)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))]
+ 105/4 *β^4*Rocs(POIo)^4 *Vons(PART)*Rpcs(POIo(t),t)^(7)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))]
+ 455/8 *β^6*Rocs(POIo)^5 *Vons(PART)*Rpcs(POIo(t),t)^(8)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^6*cos(Aθpc(POIo(t),t))]
 λ(Vons(PART)) *2 *Vons(PART)*Rpcs(POIo(t),t)^(3)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t)) *cos(Aθpc(POIo(t),t))]
 λ(Vons(PART)) *5 *β^2*Rocs(POIo)^1 *Vons(PART)*Rpcs(POIo(t),t)^(4)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))]
 λ(Vons(PART)) *10 *β^4*Rocs(POIo)^2 *Vons(PART)*Rpcs(POIo(t),t)^(5)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))]
)
/* collect powers of [sin(Aθpc(POIo(t),t)), Rocs(POIo)], move Rocs(POIo)into each line, extract Vons(PART) from each line /%
3c) + Q(PART)*β*Vons(PART)/c
*( + 21/2 *β^2*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(6)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))]
+ 105/4 *β^4*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(7)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^5*cos(Aθpc(POIo(t),t))]
+ 455/8 *β^6*Rocs(POIo)^6 *Rpcs(POIo(t),t)^(8)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^7*cos(Aθpc(POIo(t),t))]
 λ(Vons(PART)) *2 *Rocs(POIo)^1 *Rpcs(POIo(t),t)^(3)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^1*cos(Aθpc(POIo(t),t))]
 λ(Vons(PART)) *5 *β^2*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(4)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))]
 λ(Vons(PART)) *10 *β^4*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(5)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^5*cos(Aθpc(POIo(t),t))]
)
/* using /%
362:(mathL) β = Vons(PART)/c
3d) + Q(PART)*β^2
*( + 21/2 *β^2*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(6)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))]
+ 105/4 *β^4*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(7)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^5*cos(Aθpc(POIo(t),t))]
+ 455/8 *β^6*Rocs(POIo)^6 *Rpcs(POIo(t),t)^(8)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^7*cos(Aθpc(POIo(t),t))]
 λ(Vons(PART)) *2 *Rocs(POIo)^1 *Rpcs(POIo(t),t)^(3)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^1*cos(Aθpc(POIo(t),t))]
 λ(Vons(PART)) *5 *β^2*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(4)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))]
 λ(Vons(PART)) *10 *β^4*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(5)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^5*cos(Aθpc(POIo(t),t))]
)
/* collect powers of β /%
3e) + Q(PART)
*( + 21/2 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(6)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))]
+ 105/4 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(7)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^5*cos(Aθpc(POIo(t),t))]
+ 455/8 *β^8*Rocs(POIo)^6 *Rpcs(POIo(t),t)^(8)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^7*cos(Aθpc(POIo(t),t))]
 λ(Vons(PART)) *2 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(3)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^1*cos(Aθpc(POIo(t),t))]
 λ(Vons(PART)) *5 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(4)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))]
 λ(Vons(PART)) *10 *β^6*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(5)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^5*cos(Aθpc(POIo(t),t))]
)
/* using /%
3099:(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
3103:(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
3187:(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
3095:(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
3099:(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
3183:(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
3f) + Q(PART)
*( + 21/2 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(6)*sin(Aθpc(POIo(t),t=0))^4/4
+ 105/4 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(7)*sin(Aθpc(POIo(t),t=0))^6/6
+ 455/8 *β^8*Rocs(POIo)^6 *Rpcs(POIo(t),t)^(8)*sin(Aθpc(POIo(t),t=0))^8/8
 λ(Vons(PART)) *2 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(3)*sin(Aθpc(POIo(t),t=0))^2/2
 λ(Vons(PART)) *5 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(4)*sin(Aθpc(POIo(t),t=0))^4/4
 λ(Vons(PART)) *10 *β^6*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(5)*sin(Aθpc(POIo(t),t=0))^6/6
)
/* collect numbers /%
3g) + Q(PART)
*( + 21/8 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(6)*sin(Aθpc(POIo(t),t=0))^4
+ 35/8 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(7)*sin(Aθpc(POIo(t),t=0))^6
+ 455/64*β^8*Rocs(POIo)^6 *Rpcs(POIo(t),t)^(8)*sin(Aθpc(POIo(t),t=0))^8
 λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(3)*sin(Aθpc(POIo(t),t=0))^2
 λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(4)*sin(Aθpc(POIo(t),t=0))^4
 λ(Vons(PART)) *5/3 *β^6*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(5)*sin(Aθpc(POIo(t),t=0))^6
)
/* insert (3g) in place of (3a), this can be used for t=0 form /%
4) EIods(POIo,t,4th stage)
= + K_1st
+ Q(PART)
*( + 21/8 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(6)*sin(Aθpc(POIo(t),t=0))^4
+ 35/8 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(7)*sin(Aθpc(POIo(t),t=0))^6
+ 455/64*β^8*Rocs(POIo)^6 *Rpcs(POIo(t),t)^(8)*sin(Aθpc(POIo(t),t=0))^8
 λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(3)*sin(Aθpc(POIo(t),t=0))^2
 λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(4)*sin(Aθpc(POIo(t),t=0))^4
 λ(Vons(PART)) *5/3 *β^6*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(5)*sin(Aθpc(POIo(t),t=0))^6
)
+ f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}}}
/* drop "=0" qualifier to get differentiable form /%
/* using /%
1108:(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)
)
5) 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
 λ(Vons(PART)) *Rpcs(POIo(t),t)^(2)
)
+ Q(PART)
*( + 21/8 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(6)*sin(Aθpc(POIo(t),t=0))^4
+ 35/8 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(7)*sin(Aθpc(POIo(t),t=0))^6
+ 455/64*β^8*Rocs(POIo)^6 *Rpcs(POIo(t),t)^(8)*sin(Aθpc(POIo(t),t=0))^8
 λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(3)*sin(Aθpc(POIo(t),t=0))^2
 λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(4)*sin(Aθpc(POIo(t),t=0))^4
 λ(Vons(PART)) *5/3 *β^6*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(5)*sin(Aθpc(POIo(t),t=0))^6
)
+ f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}}}
6) EIods(POIo,t,4th stage)
= + Q(PART)
*( + 3/2 *β^2*Rocs(POIo)^3 *Rpcs(POIo(t),t=0)^(5)*sin(Aθpc(POIo(t),t))^2
+ 21/8 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t=0)^(6)*sin(Aθpc(POIo(t),t=0))^4
+ 35/8 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t=0)^(7)*sin(Aθpc(POIo(t),t=0))^6
+ 455/64*β^8*Rocs(POIo)^6 *Rpcs(POIo(t),t=0)^(8)*sin(Aθpc(POIo(t),t=0))^8
 λ(Vons(PART)) *1 *Rpcs(POIo(t),t=0)^(2)
 λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t=0)^(3)*sin(Aθpc(POIo(t),t=0))^2
 λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t=0)^(4)*sin(Aθpc(POIo(t),t=0))^4
 λ(Vons(PART)) *5/3 *β^6*Rocs(POIo)^3 *Rpcs(POIo(t),t=0)^(5)*sin(Aθpc(POIo(t),t=0))^6
)
+ f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}}}
/* remove "=0" from t to get :
(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))}}}}
(endMath)
/* post[differentiation, integration] form /%
/* HIGHLY restrictive conditions : t=0, RFo(t=0) = Rfp(t=0), Rocs(POIo) = Rpcs(POIo(t),t=0) /%
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)
/* use this in (6), careful to put t=0 /%
7) EIods(POIo,t,4th stage)
= + Q(PART)
*( + 3/2 *β^2 *Rpcs(POIo(t),t=0)^(2)*sin(Aθpc(POIo(t),t=0))^2
+ 21/8 *β^4 *Rpcs(POIo(t),t=0)^(2)*sin(Aθpc(POIo(t),t=0))^4
+ 35/8 *β^6 *Rpcs(POIo(t),t=0)^(2)*sin(Aθpc(POIo(t),t=0))^6
+ 455/64*β^8 *Rpcs(POIo(t),t=0)^(2)*sin(Aθpc(POIo(t),t=0))^8
 λ(Vons(PART)) *1 *Rpcs(POIo(t),t=0)^(2)
 λ(Vons(PART)) *1 *β^2 *Rpcs(POIo(t),t=0)^(2)*sin(Aθpc(POIo(t),t=0))^2
 λ(Vons(PART)) *5/4 *β^4 *Rpcs(POIo(t),t=0)^(2)*sin(Aθpc(POIo(t),t=0))^4
 λ(Vons(PART)) *5/3 *β^6 *Rpcs(POIo(t),t=0)^(2)*sin(Aθpc(POIo(t),t=0))^6
)
= + Q(PART)*Rpcs(POIo(t),t=0)^(2)
*( + 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
 λ(Vons(PART)) *1
 λ(Vons(PART)) *1 *β^2 *sin(Aθpc(POIo(t),t=0))^2
 λ(Vons(PART)) *5/4 *β^4 *sin(Aθpc(POIo(t),t=0))^4
 λ(Vons(PART)) *5/3 *β^6 *sin(Aθpc(POIo(t),t=0))^6
)
/* using /%
1174:(mathH) E0pds(POIp) = Q(PART)/Rpcs(POIp)^2
8) EIods(POIo,t,4th stage)
= + E0pds(POIp) *3/2 *β^2 *sin(Aθpc(POIo(t),t=0))^2
+ E0pds(POIp) *21/8 *β^4 *sin(Aθpc(POIo(t),t=0))^4
+ E0pds(POIp) *35/8 *β^6 *sin(Aθpc(POIo(t),t=0))^6
+ E0pds(POIp) *455/64 *β^8 *sin(Aθpc(POIo(t),t=0))^8
 E0pds(POIp)*λ(Vons(PART)) *1
 E0pds(POIp)*λ(Vons(PART)) *1 *β^2 *sin(Aθpc(POIo(t),t=0))^2
 E0pds(POIp)*λ(Vons(PART)) *5/4 *β^4 *sin(Aθpc(POIo(t),t=0))^4
 E0pds(POIp)*λ(Vons(PART)) *5/3 *β^6 *sin(Aθpc(POIo(t),t=0))^6
(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 + 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}
(endMath)
12Oct2019 This isn't working  the numbers are blowing up!! Seems to be missing negative terms...
However, at least the powers of the terms are OK now, which is a HUGE improvement!
/*_file_insert_path "$d_Lucas""relativistic factor, restrictive conditions.txt"
# enddoc
/*_endCmd
/**********************************************************
/*> (438) Binomial_expansion_for_E0_terms
23Sep2015 start 1st attempt
/*/*$ cat >>"$p_augmented" "$d_augment""04_38 work.txt"
/* Howell  do binomial expansion of :
/$ (1  β^2*sin(θ)^2)^(3/2)
Binomial series  Kreyszig1972 p52h0.0 Equation (2)
/$ (1 + z)^(m) = sum(n = 0 to ∞: choose(  m,n)*z^n)
/* where choose() is the choose operation
/$ (1 + z)^(m)
= 1  m*z
+ m*(m + 1)/2!*z^2
 m*(m + 1)*(m + 2)/3!*z^3
+ ...
/* Substituting for z= (β^2*sin^2(O)), m=3/2
/$ (1  β^2*sin(θ)^2)^(3/2)
= 1  3/2*(β^2*sin(θ)^2) + 3/2*(3/2 + 1)/2!*(β^2*sin(θ)^2)^2  3/2*(3/2 + 1)*(3/2 + 2)/3!*(β^2*sin(θ)^2)^3 + ...
= 1 + 3/2 *β^2*sin(θ)^2  15/8 * β^4*sin(θ)^4 + 105/48 * β^6*sin(θ)^6 + ...
= 1 + 3/2 *β^2*sin(θ)^2  15/8 * β^4*sin(θ)^4 + 35/16 * β^6*sin(θ)^6 + ...
/* 20Aug2019 correction  Above I had put in Lucas's factors, NOT my binomial series results!
Corrected version :
Substituting for z= (β^2*sin^2(O)), m=3/2
/$ (1  β^2*sin(θ)^2)^(3/2)
= 1 + 3/2*β^2*sin(θ)^2 + 3/8*β^4*sin(θ)^4  1/16*β^6*sin(θ)^6 + 3/128*β^8*sin(θ)^8  3/256*β^10*sin(θ)^10 + ...
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
04_38
Binomial_expansion_for_E0_terms
/$L (1  β^2*sin(θ)^2)^(3/2) = 1 + 3/2*β^2*sin(θ)^2 + 15/8*β^4*sin(θ)^4 + 35/16*β^6*sin(θ)^6 + ...
/%H (1  β^2*sin(θ)^2)^(3/2) = 1 + 3/2*β^2*sin(θ)^2 + 3/8*β^4*sin(θ)^4  1/16*β^6*sin(θ)^6 + 3/128*β^8*sin(θ)^8  3/256*β^10*sin(θ)^10 + ...
/* WRONG!  Lucas has the wrong coefifficients for a binomial series!
/*_endCmd
/**********************************************************
/*> (439) E(r,v) for constant velocity, nonpoint charge, observer reference frame
/*/*$ cat >>"$p_augmented" "$d_augment""04_39 work.txt"
Lucas
Gausss electrostatic Law from (411)()
/$ ET(r,v)
= E0(r) + Ei(r,v)
= E0(r)*(1  λ(v))/(1  β^2*sin(θ)^2)^(3/2)
/*+
First approach  using Lucas's (437)
Howell  Lucas's result for 437 Er_second_iteration
/$ ET(r,v)
= E0(r) *(1 + 3/2*β^2*sin(θ)^2 + 15/8*β^4*sin(θ)^4 + 35/16*β^6*sin(θ)^6 + ...)
 E0(r)*λ(v)*(1 + 3/2*β^2*sin(θ)^2 + 15/8*β^4*sin(θ)^4 + 35/16*β^6*sin(θ)^6 + ...)
/* Using the binomial expansion result of (438) :
04_38 Binomial_expansion_for_E0_terms
/$ (1  β^2*sin(θ)^2)^(3/2) = 1 + 3/2*β^2*sin(θ)^2 + 15/8*β^4*sin(θ)^4 + 35/16*β^6*sin(θ)^6 + ...
ET(r,v)
= E0(r) *(1  β^2*sin(θ)^2)^(3/2)
 E0(r)*λ(v)*(1  β^2*sin(θ)^2)^(3/2)
ET(r,v) = (1  λ(v))*E0(r)/(1  β^2*sin(θ)^2)^(3/2)
/* so this works well starting with Lucas's (437), but the latter may have incorrectly retained the E0*lambda term!?
/*+
Second approach  using Howells (437 iteration4 26Jun2016)
I did NOT get the same kind of result as Lucas  mine is more like Newtons binomial expansion, than the binomial series that is used in the text.
Isaac Newton :
/$ (1  x^2)^(3/2) = 1  3/2*x^2 + 3/8*x^4 + 1/16*x^6 ...
/* Howell 4th iteration :
/$ ET = + E0 *(1 + 3/2*(β*sin(θ))^2 + 3/4 *(β*sin(θ))^4 + 1/4 *(β*sin(θ))^6 + 1/8 *(β*sin(θ))^8 + 3/32 *(β*sin(θ))^10 )
 E0*λ(v)*( + 1/6 *(β*sin(θ))^6 + 1/8 *(β*sin(θ))^8 )
/* 26Jun2016 Further iterations will PERHAPS reduce the value of coefficents beyond the "betaSin^4" term, but it appears that the coefficient of the "betaSin^4" term itself WONT reduce with further iterations (?).
Binomial expansion of
/$ (1  (β*sin(θ))^2)^(3/2)
/* would give :
/% ETods(POIo,t=0)
= + E0ods(POIo,t=0)*(1 + 3/2*β*sin(Aθoc(POIo))^2 + 3/8 *β*sin(Aθoc(POIo))^4  1/16 *β*sin(Aθoc(POIo))^6 + 3/128*β*sin(Aθoc(POIo))^8  3/256*β*sin(Aθoc(POIo))^10 + 7/1024*β*sin(Aθoc(POIo))^12 + ... )
= + E0ods(POIo,t=0)*(1  3/2*β^2*sin(Aθpc(POIo(t),t=0))^2)^(3/2)
/* I assume that the E0*L(v)* term disappears with iterations, and that Howells 4th iteration is "immature" and probably harbours errors.
Therefore, at this time, just take the binomial expansion instead :
/% ETods(POIo,t=0)
= E0ods(POIo,t=0)*(1  3/2*β^2*sin(Aθpc(POIo(t),t=0))^2)^(3/2)
= E0ods(POIo,t=0)*(1 + 3/2*β*sin(Aθoc(POIo))^2 + 3/8 *β*sin(Aθoc(POIo))^4  1/16 *β*sin(Aθoc(POIo))^6 + 3/128*β*sin(Aθoc(POIo))^8  3/256*β*sin(Aθoc(POIo))^10 + 7/1024*β*sin(Aθoc(POIo))^12 + ... )
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
04_39
/* E(r,v) for constant velocity, nonpoint charge, observer reference frame
/$L ET(r,v) = (1  λ(v))*E0(r)/(1  β^2*sin(θ)^2)^(3/2)
/$H ET(r,v) = (1  λ(v))*E0(r)/(1  β^2*sin(θ)^2)^(3/2)
/* WARNING : simple using Lucas's (437), but is this incorrect? Second approach using Newtons expansion shown as well.
In the second approach : (1  L(v)) does NOT appear!!
/% ETods(POIo,t=0) = E0ods(POIo,t=0)*(1  3/2*β^2*sin(Aθpc(POIo(t),t=0))^2)^(3/2)
/*_endCmd
/**********************************************************
/*> (440) Gauss_Electrostatic_Law
/* Lucas  take from Jackson1999 p27
/$ ∬[∂(Area): ET(r)nh) = 4*π*q
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
04_40
L(v) expression for Gauss law for electric charge
/$L 4*π*q = ∬[∂(Area): ET(r)nh)
/% 4*π*Q(particle) = ∬[∂(Area)´: ETodv(POIo,t)RNpch)
/* OK  no need to do as it can be found from Jackson1999, standard formula
/**********************************************************
/*> (441) L(v) expression for Gauss law for electric charge
/*/*$ cat >>"$p_augmented" "$d_augment""04_41 work.txt"
Lucas "... using a spherical surface centered about the charge
distribution q with spherical coordinates and noting that E and n
will then be in the same direction ..."
/$ 4*π*q = ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: E0(r)*r*sin(θ)*(1  λ(v))/(1  β^2*sin(θ)^2)^(3/2)))
= 4*π*q*(1  λ(v))/(1  β^2)
/$ 4*π*q = ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: E0(r)*r*sin(θ)*(1  λ(v))/(1  β^2*sin(θ)^2)^(3/2)))
= 4*π*q*(1  λ(v))/(1  β^2)
/* Howell  Collecting Lucas's result for (439) and (440)
04_39 E(r,v) for constant velocity, nonpoint charge, observer reference frame
/$ ET(r,v) = (1  λ(v))*E0(r)/(1  β^2*sin(θ)^2)^(3/2)
/* 04_40 L(v) expression for Gauss law for electric charge
/$ 4*π*q = ∬[∂(Area): ET(r)nh)
/* Expressing (440) in terms of constant r, variable [O,P] to integrate
.over.the spherical surface :
/$1) 4*π*q
= ∬[∂(Area): ET(r)nh)
= ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: ET(r)nh)
/* Subbing (439) into (1)  BUT Lucas has extra "r*sinO" term
***Where did that extra term come from?  may be a hint for (437) etc!!!!
/$ 4*π*q
= ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: (1  λ(v))*E0(r)/(1  β^2*sin(θ)^2)^(3/2)))
= (1  λ(v))*E0(r) *∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: (1  β^2*sin(θ)^2)^(  3/2)))
/* As there is no change with P :
/$ = (1  λ(v))*E0(r) *∫[∂(φ),0 to 2*π: φ)*∫[∂(θ),0 to π: (1  β^2*sin(θ)^2)^(  3/2)))
1a) = (1  λ(v))*E0(r)*2*π *∫[∂(θ),0 to π: (1  β^2*sin(θ)^2)^(  3/2)))
/* To assess the integral in (1a), expand the series as per (438)
/$ (1  β^2*sin(θ)^2)^(3/2) = (1 + 3/2*β^2*sin(θ)^2 + 15/8*β^4*sin(θ)^4 + 35/16*β^6*sin(θ)^6 + ...)
/* So
/$ ∫[∂(θ),0 to π: (1  β^2*sin(θ)^2)^(  3/2)))
= ∫[∂(θ),0 to π: (1 + 3/2*β^2*sin(θ)^2 + 15/8*β^4*sin(θ)^4 + 35/16*β^6*sin(θ)^6 + ...)
= ∫[∂(θ),0 to π: 1)
∫[∂(θ),0 to π: 3/2*β^2*sin(θ)^2)
∫[∂(θ),0 to π: 15/8*β^4*sin(θ)^4)
∫[∂(θ),0 to π: 35/16*β^6*sin(θ)^6)
∫[∂(θ),0 to π: + ...)
/* Ignore the last term, integrate the others
= π
/$1a1) 3/2 *β^2*∫[∂(θ),0 to π: sin(θ)^2)
1a2) 15/8 *β^4*∫[∂(θ),0 to π: sin(θ)^4)
1a3) 35/16*β^6*∫[∂(θ),0 to π: sin(θ)^6)
/* Using Howell  Variables, notations, styles for Bill Lucas, Universal Force.odt
Recurring integrals from http://integraltable.com
/$1a1a) ∫[∂(θ),0 to Of: sin(θ)^2) = O/2  sin(2*O)/4
1a2a) ∫[∂(θ),0 to Of: sin^3(O)) = 3*cos(θ)/4 + cos(3*O)/12
/* This is NOT going to work!! From where did Lucas get the extra "r*sinO" term?
/* Now looking at
/$ ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: ET*(1  λ(v))*r*sin(O)/(1  β^2*sin(θ)^2)^(3/2)]}
/* [ET, L(v), b, r] are pseudoconstants wrt dO, so they come out, note above Lucas error with integral limits, but now ET instead of E0!
Also Lucas missing a second r for dP !!???!!
Correcting original :
/$ ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: E0*(1  λ(v))*r*sin(O)/(1  β^2*sin(θ)^2)^(3/2)]*r}
∫[∂(φ),0 to 2*π: E0*(1  λ(v))*r^2*∫[∂(θ),0 to π: sin(O)/(1  β^2*sin(θ)^2)^(3/2)]}
/* Now looking at
/$ ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: ET*(1  λ(v))*r*sin(O)/(1  β^2*sin(θ)^2)^(3/2)]}
/* [ET, L(v), b, r] are pseudoconstants wrt dO, so they come out, note above Lucas error with integral limits, but now ET instead of E0!
Also Lucas missing a second r for dP !!???!!
Correcting original :
/$ ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: E0*(1  λ(v))*r*sin(O)/(1  β^2*sin(θ)^2)^(3/2)]*r}
∫[∂(φ),0 to 2*π: E0*(1  λ(v))*r^2*∫[∂(θ),0 to π: sin(O)/(1  β^2*sin(θ)^2)^(3/2)]}
/* using result above for ∫[dO, 0 to π : sinO/(1  b^2*sin^2(O))^(3/2) }
/$ = ∫[∂(φ),0 to 2*π: E0*(1  λ(v))*r^2*2/(1  β^2)}
/* E0s has constant scalar magnitude at constant r for integral
/$ = E0*(1  λ(v))*r^2 *2/(1  β^2) *∫[∂(φ),0 to 2*π: 1}
= E0*(1  λ(v))*r^2 *2/(1  β^2) *2*π
/* sub E0 = q/r^2
/$ = 4*π*Q/r^2*(1  λ(v))*r^2/(1  β^2)
= 4*π*Q*(1  λ(v))/(1  β^2)
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Howell_incomplete
04_41
L(v) expression for Gauss law for electric charge
/$ 4*π*q
= ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: E0(r)*r*sin(θ)*(1  λ(v))/(1  β^2*sin(θ)^2)^(3/2)))
= 4*π*q*(1  λ(v))/(1  β^2)
/* havent done yet
PRIORITY PROBLEM  sin,cos terms
/*_endCmd
/**********************************************************
/*> (442) Special integral with binomial series (1  b^2*sin^2(O))^(3/2)
/* Started ?Sep or Oct2015?
/*/*$ cat >>"$p_augmented" "$d_augment""04_42 work.txt"
/* Howell  starting with given expression :
/$ ∫[∂(θ),0 to 2*π: sin(θ)/(1  β^2*sin(θ)^2)^(3/2)}
/* From "Howell  Background math for Lucas Universal Force, Chapter 4.odt"
Recurring integrals from http://integraltable.com
/$ ∫[∂(θ),0 to Of: sin(θ)^2)
= O/2  sin(2*O)/4  from 0 to Of
= O ^2/2  sin(2*O)/4  from 0 to Of
= (Of^2/2  sin(2*Of)/4)  (0  0)
/* Take a look at (1  b^2*sin^2(O))^(3/2)
As a hunch, take the derivative
/$ ∂(∂(θ): (1  β^2*sin(θ)^2)^(  1/2)
= (1/2)*(1  β^2*sin(θ)^2)^(3/2)*∂(∂(θ): (1  β^2*sin(θ)^2))
= (1/2)*(1  β^2*sin(θ)^2)^(3/2)*(1)*β^2*2*sin(O)*∂(∂(θ): sin(θ))
= (1/2)*(2)*β^2*sin(O)*cos(θ)*(1  β^2*sin(θ)^2)^(3/2)
= β^2*sin(O)*cos(θ)*(1  β^2*sin(θ)^2)^(3/2)
/* 27Jun2016 Now look at original integral
As a hunch, take sin(O)*(1  b^2*sin^2(O))^(1/2)
/$ ∂[∂(θ): sin(O)*(1  β^2*sin(θ)^2)^(  1/2)]
= ∂[∂(θ): sin(O)]*(1  β^2*sin(θ)^2)^(1/2) + sin(O)*∂[∂(θ): (1  β^2*sin(θ)^2)^(  1/2)]
= cos(O) *(1  β^2*sin(θ)^2)^(1/2) + sin(O)*[ β^2*sin(O) *cos(θ)*(1  β^2*sin(θ)^2)^(3/2) ]
= cos(O) *(1  β^2*sin(θ)^2)^(1/2) + β^2*sin(O)^2*cos(θ)*(1  β^2*sin(θ)^2)^(3/2)
= [ cos(O) *(1  β^2*sin(θ)^2)^1 + β^2*sin(O)^2*cos(θ) ] *(1  β^2*sin(θ)^2)^(3/2)
= [ cos(O)  cos(O)*β^2*sin(θ)^2)^1 + β^2*sin(O)^2*cos(θ) ] *(1  β^2*sin(θ)^2)^(3/2)
= cos(O) *[1  β^2*sin(θ)^2)^1 + β^2*sin(O)^2 ] *(1  β^2*sin(θ)^2)^(3/2)
= cos(O) *(1  β^2*sin(θ)^2)^(3/2)
/* Nuts  cos instead of sin (50% chance of wrong, and I did wrong)
Now try :
/$ ∂[∂(θ): cos(O)*(1  β^2*sin(θ)^2)^(  1/2)]
= ∂[∂(θ): cos(O)]*(1  β^2*sin(θ)^2)^(1/2) + cos(O)*∂[∂(θ): (1  β^2*sin(θ)^2)^(  1/2)]
= sin(O) *(1  β^2*sin(θ)^2)^(1/2) + cos(O)*[ β^2*sin(O) *cos(θ) *(1  β^2*sin(θ)^2)^(3/2) ]
= sin(O) *(1  β^2*sin(θ)^2)^(1/2) + β^2*sin(O) *cos(θ)^2*(1  β^2*sin(θ)^2)^(3/2)
= [ sin(O) *(1  β^2*sin(θ)^2)^1 + β^2*sin(O)*cos(θ)^2 ] *(1  β^2*sin(θ)^2)^(3/2)
= sin(O)*[1 + β^2*sin(θ)^2) + β^2 *cos(θ)^2 ] *(1  β^2*sin(θ)^2)^(3/2)
= sin(O)*[1 + β^2*(sin(θ)^2) + cos(θ)^2) ] *(1  β^2*sin(θ)^2)^(3/2)
= sin(O)*[1 + β^2 ] *(1  β^2*sin(θ)^2)^(3/2)
/* Therefore (NOTE! Lucas specifies dO, 0 to 2*π, but it must be 0 to π ??!!??
/$ ∫[∂(θ),0 to π: sin(θ)/(1  β^2*sin(θ)^2)^(3/2)}
= cos(O) *(1  β^2*sin(θ)^2)^(1/2) / (1 + β^2)  from 0 to π
= { cos(π) *(1  β^2*sin^2(π))^(1/2) / (1 + β^2) }
 { cos(0) *(1  β^2*sin^2(0))^(1/2) / (1 + β^2) }
= { (1) *(1  β^2*0 )^(1/2) / (1 + β^2) }
 { ( 1) *(1  β^2*0 )^(1/2) / (1 + β^2) }
= 2 /(1 + β^2)
= 2 / (1  β^2)
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Lucas_Typo_or_omission
04_42
Special integral with binomial series
/$ (1  β^2*sin(θ)^2)^(3/2)
/$L ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: ET*(1  λ(v))*r*sin(O)/(1  β^2*sin(θ)^2)^(3/2)]} = 4*π*Q*(1  λ(v))/(1  β^2)
∫[∂(θ),0 to 2*π: sin(θ)/(1  β^2*sin(θ)^2)^(3/2)) = 2/(1  β^2)
/* and
/$ E0(r) = q/r^2
/* so
/$L λ(v) = β^2
/$H ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: E0*(1  λ(v))*r*sin(O)/(1  β^2*sin(θ)^2)^(3/2)]*r} = 4*π*Q*(1  λ(v))/(1  β^2)
∫[∂(θ),0 to 2*π: sin(θ)/(1  β^2*sin(θ)^2)^(3/2)) = 2/(1  β^2)
/* and
/$ E0(r) = q/r^2
/* so
/$ λ(v) = β^2
/* OK  easy, but Lucas uses wrong 2*π rather than π upper limit on sinO/(1  b^2*sin^2(O))^(3/2) integral, and uses ET rather than E0, and is missing an r!
HOWEVER : my 437 is missing the lambda term  so my earlier resilts dont work with this (maybe some other twisted argument will do for me?).
/*_endCmd
/**********************************************************
/*> (4_43) E&B_fields_self_consistent
Lucas
/$ ET(r,v) = E0(r) + Ei(r,v) = E0(r)*(1  β^2)/(1  β^2*sin(θ)^2)^(3/2)
Bi(r,v) = v/cET(r,v)
/* Howell  from (442)
/$ λ(v) = β^2
/* Substituting into (439)
04_39 E(r,v) for constant velocity, nonpoint charge, observer reference frame
/$ ET(r,v) = (1  λ(v))*E0(r)/(1  β^2*sin(θ)^2)^(3/2)
ET(r,v) = E0(r)*(1  β^2)/(1  β^2*sin(θ)^2)^(3/2)
/* From (413)
04_13
Total B magnetic flux density as induced from E0 + Ei
/$ B(r´,t´) = (v/c)[ E0(r´,t´) + Ei(r´,t´) ]
B(r´,t´) = (v/c)ET(r´,t´)
/* 29May2016  As E&B are measured at a POI they give the same result in either RFp or RFo for that POI.
What does matter is whether we are refering to a fixed POIo, or POIp. There is no B in the latter case.
earlier comment : Lucas drops the primes, so I will too. Also  we are dealing with a constant v
and particle/system frame of reference, and (from 04_13 :
***where B0 is dropped as it "... is not electrostatic in nature ..." (p67h0.4)
/$ B(r,v,t) = B0(r,t) + Bi(r,v,t) = Bi(r,v,t)
/* so
/$ B(r´,t´) = (v/c)ET(r´,t´)
/* Lucas does NOT substitute the expression (439) for E(r,v)
AND  he includes the ELECTROSTATIC E????
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
04_43
E&B_fields_self_consistent
/$L ET(r,v) = E0(r)*(1  β^2)/(1  β^2*sin(θ)^2)^(3/2)
Bi(r,v) = (v/c)ET(r,v)
/$H
/% ETodv(POIo,t) = E0odv(POIo,t)*(1  β^2)/(1  β^2*sin^2(Aθpc(POIo(t),t)))^(3/2)
BIodv(POIo,t) = Vons(PART)/cEIodv(POIo,t)
/* OK  simple, although I think primes are needed to denote particle reference frame (RFp) for angle theta (Op).
This does not apply for E,B vectors, for which the primes are unimportant  the direction is the same in both (RFo) and (Rfp).
/**********************************************************
/*> (444) F_total by moving charge distribution on a test charge q'
/*/*$ cat >>"$p_augmented" "$d_augment""04_44 work.txt"
Lucas
/$ F(r,v) = q*{ET(r,v) + (v/c)Bi(r,v)}
= q* E0(r) * (1  β^2) /(1  β^2*sin(θ´)^2)^(3/2) *[(1  β^2 + β^2*cos(θ´)^2)*r  (β•r)*r(rβ)]
= q* E0(r) * (1  β^2) /(1  β^2*sin(θ´)^2)^(3/2) *[(1  β^2*sin(θ´)^2)*r  (β•r)*r(rβ)]
= q* E0(r) * (1  β^2) /(1  β^2*sin(θ´)^2)^(1/2)  q*E0(r)* (1  β^2) /(1  β^2*sin(θ´)^2)^(3/2) *(β•r)*r(rβ)
/* Can one say? :
where the first term is conventional (MaxwellianRelativity)
and the second is NEW from Lucas's Universal Force
/* Howell  starting with (426)
Derived Lorentz Force F_L(const v : q,E,Bi)
/$ F(r´,t´) = q*ET(r  v*t,t) + q/c*[vBi(r  v*t,t)]
/* Lucas puts into observer frame of reference
/$1) F(r,v) = q*ET(r,v) + q/c*[vBi(r,v)]
/* Using (443) E&B_fields_self_consistent
/$1a) ET(r,v) = E0(r)*(1  β^2)/(1  β^2*sin(θ´)^2)^(3/2)
Bi(r,v) = (v/c)ET(r,v)
/* Putting (1a) terms into (1) :
/$ F(r,v)
= q*E0(r)*(1  β^2)/(1  β^2*sin(θ´)^2)^(3/2) + q/c*{ v[ v/cE0(r)*(1  β^2)/(1  β^2*sin(θ´)^2)^(3/2) ] }
= q *(1  β^2)/(1  β^2*sin(θ)^2)^(3/2) *[ E0(r) + 1/c*v[ v/cE0(r) ]
2) = q *(1  β^2)/(1  β^2*sin(θ)^2)^(3/2) *[ E0(r) + v/c[ v/cE0(r) ]
/* Consider (2)  the expression (v/c)[(v/c)E0(r)]
Lucas provides a corresponding vector identity in (445) below (this is verified later) :
04_45 Vector identities for Lorentz Force derivation
/$3) (v/c)[(v/c)E0(r,v)] = (v/c)•E0(r)*[(v•rh)*rh/c  rh(rhv)/c]  (vs/c)^2*E0(r)
/* Subbing (3) into (2) :
/$ F(r,v)
= q*(1  β^2)/(1  β^2*sin(θ´)^2)^(3/2) *{ E0(r) + (v/c)•E0(r)*[(v•rh)*rh/c  rh(rhv)/c]  (vs/c)^2*E0(r) }
4) = q*(1  β^2)/(1  β^2*sin(θ´)^2)^(3/2) *{ E0(r) + (v/c)•E0(r)* (v•rh)*rh/c  (v/c)•E0(r)*rh(rhv)/c  (vs/c)^2*E0(r) }
/* looking at term :
/$ (v /c)•E0 (r) *(v •rh )*rh/c
= (v /c)•E0 (r) *(v /c •rh )*rh
= (vs/c)*E0s(r)*cos(θ´)*(vs/c*rh*cos(θ´))*rh
= (vs/c) *cos(θ´)*(vs/c*1 *cos(θ´))*rh*E0s(r)
= (vs/c)^2*cos(θ´)^2*rh*E0s(r)
/* but
/$ rh*E0s(r) = E0(r), vs/c = β
/* so
5) ???? = b^2*cos(Op)^2*E0(r)
/* looking at term :
/$ (v/c)• E0 (r)*rh(rh v)/c
= (v/c)•rh*E0s(r)*rh(rh(v /c)) ET(r,v) = E0(r)*(1  β^2)/(1  β^2*sin(θ)^2)^(3/2)
/* Now setting b_v = b*v_h = v/c
/$6) = E0s(r)*(b_v•rh)*rh(rhb_v)
/* subbing (5) and (6) into (4) :
/$4) F(r,v)
= q*(1  β^2)/(1  β^2*sin(θ´)^2)^(3/2)*{ E0(r) + (v/c)•E0(r)* (v•rh)*rh/c  (v/c)•E0(r)*rh(rhv)/c  (vs/c)^2*E0(r) }
7) = q*(1  β^2)/(1  β^2*sin(θ´)^2)^(3/2)*{ E0(r) + β^2*cos(θ´)^2*E0(r)  E0s(r)*(b_v•rh)*rh(rhb_v)  (vs/c)^2*E0(r) }
/* NOTE : from (7) above, it appears that the 2nd and third expressions in Lucas (444) are incorrect intermediates
as E0(r) does NOT factor out as shown from E0s(r)*(b_v•r_h)*r_h(r_hb_v).
/* From (7), separate terms with E0(r) and E0s(r) within the curly brackets :
/$ F(r,v)
= q*(1  β^2)/(1  β^2*sin(θ´)^2)^(3/2)*{ E0(r) + β^2*cos(θ´)^2*E0(r)  E0s(r)*(b_v•rh)*rh(rhb_v)  (vs/c)^2*E0(r) }
= q*(1  β^2)/(1  β^2*sin(θ´)^2)^(3/2)*{ E0(r)*[ 1 + β^2*cos(θ´)^2  (vs/c)^2 ]  E0s(r)*(b_v•rh)*rh(rhb_v) }
/* subbing vs/c = b
/$ = q*(1  β^2)/(1  β^2*sin(θ´)^2)^(3/2)*{ E0(r)*[ 1 + β^2 *cos(θ´)^2  β^2 ]  E0s(r)*(b_v•rh)*rh(rhb_v) }
= q*(1  β^2)/(1  β^2*sin(θ´)^2)^(3/2)*{ E0(r)*[ 1 + β^2*(cos(θ´)^2  1) ]  E0s(r)*(b_v•rh)*rh(rhb_v) }
8) = q*(1  β^2)/(1  β^2*sin(θ´)^2)^(3/2)*{ E0(r)*[ 1  β^2 *sin(θ´)^2 ]  E0s(r)*(b_v•rh)*rh(rhb_v) }
/* multiply q*(1  b^2)/(1  b^2*sin^2(Op))^(3/2) into curly brackets :
/$ = q*(1  β^2)/(1  β^2*sin(θ´)^2)^(3/2)* E0(r)*[ 1  β^2 *sin(θ´)^2]  q*(1  β^2)/(1  β^2*sin(θ´)^2)^(3/2)* E0s(r)*(b_v•rh)*rh(rhb_v)
= q*(1  β^2)/(1  β^2*sin(θ´)^2)^(1/2)* E0(r)  q*(1  β^2)/(1  β^2*sin(θ´)^2)^(3/2)* E0s(r)*(b_v•rh)*rh(rhb_v)
= q*E0(r)*(1  β^2)/(1  β^2*sin(θ´)^2)^(1/2)  q*E0s(r)*(1  β^2)/(1  β^2*sin(θ´)^2)^(3/2) *(b_v•rh)*rh(rhb_v)
/* Summarizing, but writing E0s(r) = E0(r) :
/$9) F(r,v)
= q* E0(r) *(1  β^2)/(1  β^2*sin(θ´)^2)^(1/2)  q*E0(r)*(1  β^2)/(1  β^2*sin(θ´)^2)^(3/2)* (b_v•rh)*rh(rhb_v)
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Lucas_Typo_or_omission
04_44rev1 30May2016
F_electromag_total_constant_v
/$L F(r,v) = q*{ET(r,v) + (v/c)Bi(r,v)} = q *E0(r) *(1  β^2)/(1  β^2*sin^2(O))^(1/2)  q*E0(r)*(1  β^2)/(1  β^2*sin^2(O))^(3/2)*(b_v•rh)*rh(rhβ)
/$H ??????????
/% FTodv(POIo,t) = Q(particle) *E0odv(POIo,t) *(1  β^2)/(1  β^2*sin^2(Aθpc(POIo(t),t)))^(1/2)  Q(particle) *E0odv(POIo,t) *(1  β^2)/(1  β^2*sin^2(Aθpc(POIo(t),t)))^(3/2) *(beta_v•Roch(POIo))*Roch(POIo)(Roch(POIo)beta_v)]
/* where beta_v = beta*Vonv(PART)
OK  works, Problem  usage of angle Oo in observer reference frame (RFo) instead of Op in (RFp)
Oo = Op really applies ONLY when frames are coincident & aligned  here Lucas should use (RFp) primes (eg Op)
should Roch(POIo) below be Rodh(Vonv_X_Rpcv(POIo)) from file "Howell  Background math for Lucas Universal Force, Chapter 4.odt"?
/*_endCmd
/**********************************************************
/*> (445) Vector identities for Lorentz Force derivation
Lucas  "The following identities were used for Lucas04_44 "
Howell  checks from references
Lucas's expression (445) follows directly from (446)
/$ v/c[v/cE0(r,v)]
= (v/c)•E0(r)*(v/c)  (vs/c)^2*E0(r)
= (v/c)•E0(r)*[(v•rh)*rh/c  rh(rhv)/c]  (vs/c)^2*E0(r)
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
04_45
Vector identities for Lorentz Force derivation
/$L v/c[v/cE0(r,v)] = (v/c)•E0(r)*[(v•rh)*rh/c  rh(rhv)/c]  (vs/c)^2*E0(r)
/$H ???
/% Vonv(PART)/c[Vonv(PART)/cE0(r,v)] = (Vonv(PART)/c)•E0odv(POIo,t)*[(Vonv(PART)•Roch(POIo))*Roch(POIo)/c  Roch(POIo)(Roch(POIo)Vonv(PART))/c]  (Vons(PART)/c)^2*E0odv(POIo,t)
/* OK  not required as follows directly from (446)
should Roch(POIo) below be Rodh(Vonv_X_Rpcv(POIo)) from file "Howell  Background math for Lucas Universal Force, Chapter 4.odt"?
/**********************************************************
/*> (446) Vector_operations used for the Lorentz force
Lucas  "The following identities were used for Lucas04_44 "
Howell  checks from references
From Howell  Variables, notations, styles for Bill Lucas, Universal Force.odt
scalar triple products Kreyszig Section 5.9 p213216
/$1) β(c∂) = (β∂)*c  (βc)*∂
/* This takes care of the first expression
The second equation in (446) :
/$2) v = v  (vrh)*rh + (vrh)*rh
= (vrh)*rh  rh(rhv)
/* The first step is straightforward  just add and subtract the same term :
/$ v = v  (vrh)*rh + (vr)*r
/* Reverseprocessing the 2nd step, using (1)
/$ rh(rhv) = (rhv)*rh  (rhrh)*v
/* but r_hr_h = 1 (unit vectors same direction), therefore
/$2a) rh(rhv) = (rhv)*rh  v
/* Putting (2a) into (2)
/$ v = (vrh)*rh  rh(rhv)
= (vrh)*rh  ((rhv)*rh  v)
= v
/*So this works
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
04_46
Vector_operations used for the Lorentz force
/$L A(BC) = (A•C)*B  (A•B)*C
/$H A(BC) = (A•C)*B  (A•B)*C
/* OK  very simple, from textbooks
/********************************************** ;
>>> Lucas 5  Extension of the Universal Force Law to include acceleration
/********************************************** ;
>>>>>> 5.1  Generalized electromagnetic potential U(r,v)
/**********************************************************
/*> 05_01 ;
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
05_01
?title?
same as Lucas04_44
no need  same as Lucas04_44
no need  same as Lucas04_44
/**********************************************************
/*>
Lucas
Generalized_potential_U
/$ U(r,v) = F(r,v)•r
∂[∂t: U) = F(r,v)•∂[∂t: r)  ∂[∂t: F(r,v))•r
= F(r,v)•v  ∂[∂t: F(r,v))•r
/* For stability iunder a constant force :
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
05_02
Generalized_potential_U
/$ ∂[∂t: U) = F(r,v)•v  ∂[∂t: F(r,v))•r
/* havent done yet
/* havent done yet
/**********************************************************
/*> 05_03 ;
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
05_03
?title?
/$ ∂[∂t: U(r,v)) = F(r,v)•v
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas
Generalized_potential_U
/$ U(r,v)
= q*q´/r*(1  β^2)/(1  β^2*sin(θ)^2)^(1/2)
= q*q´/r*(1  β^2)/[r^2  {r(rβ)/r^2}]^(1/2)
where β=v/c and
∂[∂t: U(r,v) =  v•F(r,v,a)
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
05_04
Generalized_potential_U
/$ U(r,v) = q*q´/r*(1  β^2)/[r^2  {r(rβ)/r^2}]^(1/2)
/* where
/$ β=v/c
/* and
/$ ∂[∂t: U(r,v) =  v•F(r,v,a)
/* havent done yet
/**********************************************************
/*>
Lucas
/$ v•F
= ∂[∂t: U(r,v))
= ∂/∂(t)[q*q´*(1  β^2)/[r^2  {r(rβ)}^2/r^2]^(1/2)]
= q*q´*{ ∂/∂(t)[(1  β^2)/[r^2  {r(rβ)}^2/r^2]^(1/2)
+ (1  β^2)/[r^2  {r(rβ)}^2/r^2]^(1/2)
* (1/2)*∂/∂(t)[r^2  {r(rβ)}^2/r^2]^(1/2)
/ [r^2  {r(rβ)}^2/r^2]^(3/2)
}
= q*q´*{ 2*(v/c)•(a/c)/[r^2  {r(rβ)}^2/r^2]^(1/2)
+ 1/2*(1  β^2)/[2*r•v  {r(rβ)}^2*2*r•v/r^4]
/ [r^2  {r(rβ)}^2/r^2]^(3/2)
}
+ q*q´*{ 1/2*(1  β^2)*2*{r(rβ)}/r^2 • ∂/∂(t){r*(r•β)  β*(r•r)}
/ [r^2  {r(rβ)}^2/r^2]^(3/2)
}
= q*q´*{ 2*(v/c)•(a/c)/[r^2  {r(rβ)}^2/r^2]^(1/2)
+ (1  β^2)*r•v* [1  {r(rβ)}^2/r^4]
/ [r^2  {r(rβ)}^2/r^2]^(3/2)
}
+ q*q´ *(1  β^2) *{r(rβ)} /r^2
•{v*(r•β) + r*v^2/c + r*(r•a/c)  a/c*r^2  2*β*(v•r)}
/ [r^2  {r(rβ)}^2/r^2]^(3/2)
}
= q*q´/r^2*{ (1  β^2)*v•r + 2*r^2/c^2*v•a }
/ [r^2  {r(rβ)}^2/r^2]^(1/2)
+ q*q´ *(1  β^2) *??x?? *{r(rβ)}
•{v/c*(v•r) + r*v^2/c} + {r*(r•β)  β*r^2}•r(ra/c)
}
/ [r^2  {r(rβ)}^2/r^2]^(3/2)
= q*q´/r^2*{ (1  β^2)*v•r + 2*r^2/c^2*v•a }
/ [r^2  {r(rβ)}^2/r^2]^(1/2)
 q*q´ *(1  β^2)*
{ r(rβ)•v/c*(v•r) + r^2*r(ra/c)•β }
/ [r^2  {r(rβ)}^2/r^2]^(3/2)
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
05_05_pre
?title?
/$ ( v•F
= q*q´/r^2*{ (1  β^2)*v•r + 2*r^2/c^2*v•a }
/ [r^2  {r(rβ)}^2/r^2]^(1/2)
 q*q´ *(1  β^2)*
{ r(rβ)•v/c*(v•r) + r^2*r(ra/c)•β }
/ [r^2  {r(rβ)}^2/r^2]^(3/2)
)
/* havent done yet
/* havent done yet
/**********************************************************
/*>
p79h0.2 the following vector identities were used
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
05_06
?title?
/$ A(BC)  (A•C)*B  (A•B)*C
/$L r•r(rβ) = (r•β)*(r•r)  (r•r)*(r•β) = 0
/$H??? r•r(ra) = (r•a)*(r•r)  (r•r)*(r•a) = 0
/* havent done yet
havent done yet
/**********************************************************
/*>
28Aug2015  look at this later. Might help to get rid of "extra" "r"!?!?
H5_07 := Universal_ED_force_with_acceleration_simplified :=
/$ F(r,v,a) =
= q*q´/r^2*(1  β^2)*r + 2*r^2/c^2*a }
/ [r^2  {r(rβ)}^2/r^2]^(1/2)
 (1  β^2)* ???? do later ????
{ (β•r)*r(rβ) + (r•r)*r(ra/c^2) }
/ [r^2  {r(rβ)}^2/r^2]^(3/2)
}
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
05_07
Universal_ED_force_with_acceleration
/$L ( F(r,v,a) =
= q*q´/r^2*
[ + { (1  β^2)*r + 2*r^2/c^2*a }
/ [r^2  {r(rβ)}^2/r^2]^(1/2)
 (1  β^2)*
{ (β•r)*r(rβ) + (r•r)*r(ra/c^2) }
/ [r^2  {r(rβ)}^2/r^2]^(3/2)
]
)
/* havent done yet
havent done yet
/**********************************************************
/*> 05_08 ;
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
05_08
Phipps_ED_force_relativistic_transverse
Wesleys_ED_force_relativistic_circular
/$ F(r,v,a: rperpendicular to v,a = 0)
= q*q´*rh/r^2*[(1  β^2)/(1  β^2*sin(θ)^2)^(1/2) ](@O=π/2)
≈ q*q´*rh/r^2* (1  β^2)^(1/2)
/* havent done yet
havent done yet
/**********************************************************
/*>
p80h0.8 In the nonrelativistic limit v<
Lucas
F_Lienard_Wichert_electric_field_v_muchless_c
/$ ≈ q*E_v_muchless_c
≈ q*E0 + q*(Ea + Erad)
where q*(Ea + Erad) ≈ q*q´/c^2/r*[2*a + r*(r•a)  a]
therefore F_Lienard_Wichert_v_muchless_c
= q*q´*r/rs^3 + q*q´/c^2/r*[2*a + r*(r•a)  a]
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
05_10
/$ F_Lienard_Wichert_electric_field_v_muchless_c
( F_Lienard_Wichert v<>>>>> 5.2  Acceleration fields and radiation ;
/**********************************************************
/*> 05_11 ;
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
05_11
Acceleration fields and radiation
/$ FE = q*ET(r) = q*[E0(r) + Ei(r) ]
FM = v/c[B0(r) + Bi(r)]
/* havent done yet
/* havent done yet
/**********************************************************
/*> 05_12 ;
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
05_12
Acceleration fields and radiation
/$ Ei(r) = Ev(r) + Ea(r) + Erad(r) + ...
Bi(r) = Bv(r) + Ba(r) + Brad(r) + ...
/* havent done yet
/* havent done yet
/**********************************************************
/*> 05_13 ;
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
05_13
Acceleration fields and radiation
/$ F = F_E0 + F_Ev + F_Ea + ...
+ F_B0 + F_Bv + F_Ba + ...
+ F_rad
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas
/$ F_E0 + F_Ev
= q*q´*r/rs^3*(1  β^2) /(1  β^2*sin(θ)^2)^(1/2)
= q´*(E0 + Ev)
F_B0 + F_Bv =
= q*q´*r/rs^3*(1  β^2)*β^2*sin(θ)^2 /(1  β^2*sin(θ)^2)^(1/2)
+ q*q´ /rs^4*(1  β^2)*(β•r)*r(rβ)/(1  β^2*sin(θ)^2)^(3/2)
/*
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
05_14
Acceleration fields and radiation
/$ F_E0 + F_Ev = q´*(E0 + Ev)
F_B0 + F_Bv =
= q*q´*r/rs^3*(1  β^2)* β^2*sin(θ)^2 /(1  β^2*sin(θ)^2)^(1/2)
+ q*q´ /rs^4*(1  β^2)*(β•r)*r(rβ)/(1  β^2*sin(θ)^2)^(3/2)
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Poynting vector S  radiation from charged particle acceleration
Poynting_radiation_flux_from_charged_particle_acceleration
/$ S = c/4/π*Erad^2*r
= q^2/4/π/c^3*r(ra)^2/r^2*(1  β^2)^2/(1  β^2*sin(θ)^2)^3
/*
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
05_15
Poynting_radiation_flux_from_charged_particle_acceleration
/$ S = q^2/4/π/c^3*r(ra)^2/r^2*(1  β^2)^2/(1  β^2*sin(θ)^2)^3
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas
Poynting_radiation_per_solid_angle
/$ ∂/dΩ(φ)
= r^2*S
= q^2/4/π/c^3*r(ra)^2 *(1  β^2)^2/(1  β^2*sin(θ)^2)^3
/* Op is angle between r & a
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
05_16
Poynting_radiation_per_solid_angle
/$ ∂/dΩ(φ) = q^2/4/π/c^3*r(ra)^2*(1  β^2)^2/(1  β^2*sin(θ)^2)^3
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Op is angle between r & a
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
05_17
Poynting_radiation_per_solid_angle_nonrelativistic
/$ ∂/dΩ(φ)(@v<
Lucas
Power_ Larmor_radiation_total_nonrelativistic
/$ P_ Larmor_radiation_nonRel
= ∫[dPp,0 to 2*π:
∫[∂(θ),  1 to 1: q^2/4/π/c^3*a^2*(1  cos(θ´)^2)]]
= 2/3/c^3*q^2*a^2
/*
p82h0.5 For the relativistic case of a circular accelerator where v is
perpendicular to r such that sinO = 1 or r•v = 0 and a is perpendicular
to b = v/c
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
05_18
Power_ Larmor_radiation_total_nonrelativistic
/$ P_ Larmor_radiation_nonRel = 2/3/c^3*q^2*a^2
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas
/* P_Lienard_circular_accelerator_relativistic
/$ ∂/dΩ(P_Lienard)
= q^2/4/π/c^3 *a^2*sin(θ´)^2*(1  β^2)^2/(1  β^2*sin(θ)^2)^3
= q^2/4/π/c^3*γ^4*a^2*sin(θ´)^2*(1  β^2)
/* where γ = ???????????
/$ P_Lienard = 2/3*q^2 /c^3*γ^4*[a^2  (βa)^2]
/* using the (518) integral
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
05_19
P_Lienard_circular_accelerator_relativistic
/$ P_Lienard = 2/3*q^2/c^3*γ^4*[a^2  (βa)^2]
/* havent done yet
/* havent done yet
/*********************
>>> Lucas 6  Extension of the Universal Force Law to include radiation reaction da/dt
/********************************************** ;
>>>>>> 6.3  Derivation of nonrelativistic radiation reaction force ;
U  electrodynamic potential energy for constant velocity
/**********************************************************
/*>
Lucas06_01 := Lucas05_04 ;
p85h0.3 Note that
/$ ∂[∂t: U(r,v) =  v•F(r,v,a)
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
06_01
Generalized_potential_U
same as Lucas05_04
no need same as Lucas05_04
no need same as Lucas05_04
/**********************************************************
/*>
Lucas06_02 := Lucas05_07 ;
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
06_02
Universal_ED_force_with_acceleration
same as Lucas05_07
no need  same as Lucas05_07
no need  same as Lucas05_07
/**********************************************************
/*>
LucasWork_done_on_particle_by_EDF_to_emit_radiation_nonrelativistic :=
/$ ∫[∂(t),τ1 to τ2: F_rad•v)
= ∫[∂(t),τ1 to τ2: Lucas05_18Larmor_radiation_total_nonrelativistic)
= ∫[∂(t),τ1 to τ2: 2/3/c^3*q^2*a^2)
= ∫[∂(t),τ1 to τ2: 2/3/c^3*q^2*(∂[∂t: v))^2)
/*
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
06_03
/$ Work_done_on_particle_by_EDF_to_emit_radiation_nonrelativistic
∫[∂(t),τ1 to τ2: F_rad•v)
= ∫[∂(t),τ1 to τ2: 2/3/c^3*q^2*(∂[∂t: v))^2)
/* havent done yet
/* havent done yet
/**********************************************************
/*>
integrate by parts
Lucas06_04 := Work_done_on_particle_by_EDF_to_emit_radiation_nonrelativistic
/$ Work_done_on_particle_by_EDF_to_emit_radiation_nonrelativistic
= ∫[∂(t),τ1 to τ2: F_rad•v)
= ∫[∂(t),τ1 to τ2: 2/3/c^3*q^2*(∂[∂t: v))^2)
= 2/3/c^3*q^2*
[  [∂[∂t: v)•v]τ2
+ [∂[∂t: v)•v]τ1
+ ∫[∂(t),τ1 to τ2: d2/dt2(v)•v)
]
= 0 + 2/3/c^3*q^2*∫[∂(t),τ1 to τ2: d2/dt2(v)•v)
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
06_04
/$ Work_done_on_particle_by_EDF_to_emit_radiation_nonrelativistic
= ∫[∂(t),τ1 to τ2: F_rad•v)
= 0 + 2/3/c^3*q^2*∫[∂(t),τ1 to τ2: d2/dt2(v)•v)
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Comparing the integrands of the left and right hand sides of the equation,
we can identify the experimentallyconfirmed nonrelativistic radiation
reaction force :
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
06_05
F_radiation_reaction_nonrelativistic
/$ F_rad = 2/3/c^3*q^2*∂[∂t: a)
/* havent done yet
/* havent done yet
/********************************************** ;
>>>>>> 6.4  Derivation of relativistic radiation reaction force ;
/**********************************************************
/*>
circular accelerator v is perpendicular to r such that sinO = 1 or r•v = 0
Lucas05_19 :=
P_Lienard_circular_accelerator_relativistic
/$ ∂/dΩ(P_Lienard)
= q^2/4/π/c^3 *a^2*sin(θ´)^2*(1  β^2)^2/(1  β^2*sin(θ)^2)^3
= q^2/4/π/c^3*γ^4*a^2*sin(θ´)^2*(1  β^2)
/* where γ = ???????????
/$ P_Lienard = 2/3*q^2 /c^3*γ^4*[a^2  (βa)^2]
/* using the (518) integral
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
05_19
P_Lienard_circular_accelerator_relativistic
/$ P_Lienard = 2/3*q^2 /c^3*γ^4*[a^2  (βa)^2]
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas06_06 :=
Work_done_on_particle_by_EDF_to_emit_radiation_relativistic
/$ ∫[∂(t),τ1 to τ2: F_rad•v)
= ∫[∂(t),τ1 to τ2: Lucas05_19: = P_Lienard_circular_accelerator_relativistic)
= ∫[∂(t),τ1 to τ2: P_rad)
/* from Lucas05_19
/$ = ∫[∂(t),τ1 to τ2: 2/3/c^3*q^2*γ^4*[a^2  (βa)^2]
/* from Lucas p86h0.75
/$ = ∫[∂(t),τ1 to τ2: 2/3/c^3*q^2*γ^2*a^2)
/* #NOTE!!! Lucas has gamma^2, not gamma^4, dropped 2nd expression!!!!
/$ = ∫[∂(t),τ1 to τ2: 2/3/c^3*q^2*γ^2*∂[∂t: v)•∂[∂t: v))
= 0 + 2/3/c^3*q^2*γ^2*∫[∂(t),τ1 to τ2: d2/dt2(v)•v)
/* NOTE : change in integral  I dont trust this!!
assuming periodicity in charged particle structure,
boundary terms in the integral by parts disappear
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
06_07
F_radiation_circular_accelerator_relativistic
/$ F_rad = 2/3/c^3*q^2*γ^2*∂[∂t: a)
/* havent done yet
/* havent done yet
/********************
>>> Lucas 7  Electrodynamic origin of gravitational forces
/********************************************** ;
>>>>>> 7.1  Introduction, Electrodynamic origin of gravitational forces ;
/**********************************************************
/*> 07_01 ;
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_01
Commonality_of_electrodynamics_and_gravity
/$ F_Coulomb = q1 *q2 /r^2
F_gravity = mg1*mg2/r^2
/* where r = r2  r1
/* havent done yet
/* havent done yet
/**********************************************************
/*> 07_02 ;
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_02
/$ Gμv = 8*π/c*G*Tμv
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas
Combined_Lorentz_force_Ampere_induction
/$ F = q*E0 + v/cBi
= q*E0 + v/c(q*v/cE0)
= q*E0*(1 + (v/c)^2)
/*
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_03
Combined_Lorentz_force_Ampere_induction
/$ F_Lorentz_AmpInductn = q*E0*(1 + (v/c)^2)
/* havent done yet
/* havent done yet
/**********************************************************
/*> 07_04 ;
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_04
From_free_electron_drift_in_conductors
/$ (v/c)^2 ≈ [ 3*10^(2)/3*10^8 ]^2 ≈ 10^(20)
/* speeds in m/sec
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas07_05 := Lucas05_07 ;
Lucas05_07 :=
Universal_ED_force_with_acceleration
/$ F(r,v,a) =
= q*q´/r^2*
[ + { (1  β^2)*r + 2*r^2/c^2*a } / [r^2  {r(rβ)}^2/r^2]^(1/2)
 (1  β^2)*{ (β•r)*r(rβ) + (r•r)*r(ra/c^2) } / [r^2  {r(rβ)}^2/r^2]^(3/2)
]
/* first acceleration term, a, gives rise to Newtons Second Law (F=m*a)
second acceleration term, ra, gives rise to absorption/emission
of electromagnetic radiation or light
For acceleration a=0 and writing out b=v/c
???????? too many "r"????
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_05
Universal_ED_force_with_acceleration
same as Lucas05_07
no need  same as Lucas05_07
no need  same as Lucas05_07
/**********************************************************
/*>
?? conversion of [r^2  {r(rb)}^2/r^2]
to (1  b^2*sin^2(O))^(3/2)
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_06
Universal_force_same_as_covariant_relativistic_Maxwell
/$ F(r,v,a) =
= + q*q´/r^2*(1  β^2)*r /(1  β^2*sin(θ)^2)^(1/2)
 q*q´/r^2*(1  β^2)*(r•v/c)*r(rv/c)/(1  β^2*sin(θ)^2)^(3/2)
/* havent done yet
/* havent done yet
/********************************************** ;
>>>>>> 7.2  Origin of gravitational forces ;
Gaussian system of units, assuming constant velocity a=0, b=v/c,
binomial expansion of radial term in (76), keep only to order b^4, and
substitute sin^2(O) = 1  cos^2(O)
/**********************************************************
/*>
Lucas
Universal ED force, no acceleration a, expanded radial term
/$ F(r,v)
= + q*q´*r/r^2 *(1  β^2)*[1 + 1/2*β^2*sin(θ)^2 + (1/2)*(3/2)/2*β^4*sin(θ)^4 + ...]
 q*q´ /r^2*(r•β)*r(rβ)*(1  β^2)*[1 + 3/2*β^2*sin(θ)^2 + (3/2)*(5/2)/2*β^4*sin(θ)^4 + ...]
= + q*q´*r/r^2 *[1  1/2*β^2  1/2*β^2*cos(θ)^2  1/8*β^4  1/4*β^4*cos(θ)^2 + 3/8*β^4*cos(θ)^4 + ...]
/*
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_07
Universal ED force, no acceleration a, expanded radial term
/$ F(r,v)
= + q*q´*r/r^2 *[1  1/2*β^2  1/2*β^2*cos(θ)^2  1/8*β^4  1/4*β^4*cos(θ)^2 + 3/8*β^4*cos(θ)^4 + ...]
 q*q´ /r^2*(r•β)*r(rβ)*[1 + 1/2*β^2  3/2*β^2*cos(θ)^2  3/8*β^4  9/4*β^4*cos(θ)^2 + 15/8*β^4*cos(θ)^4 + ...]
/* havent done yet
/* havent done yet
/**********************************************************
/*> 07_08 ;
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_08
Total force between two neutral dipoles
labels : + positive (proton),  negative (electron), dipole1, dipole2
/$ F = F(2 + ,1 + ) + F(2 + ,1  ) + F(2  ,1 + ) + F(2  ,1  )
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas07_Fig7_2 looks wrong? check later
Lucas07_09 is written incorrectly!?!?!?
P, P1, P2, O, t1, t2 are undefined!!
It seems to me that each interaction should have been specified for the integration!
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_Fig7_2
Neutral dipole geometry and frequencies, Lucas Figure 72
/$ r(1 + ,2 + ) = r(2 + )  r(1 + ), w1 = 2*π*f1, w2 = 2*π*f2
r(1  ,2 + ) = r(2 + )  r(1  )  A1*cos(w1*t + P1), A1*f1 = v1
r(1 + ,2  ) = r(1 + )  r(2  )  A2*cos(w2*t + P2), A2*f2 = v2
r(2  ,1  ) = r(2  )  r(1  )  A2*cos(w2*t + P2)  A1*cos(w1*t + P1)
/* havent done yet
/* havent done yet
/**********************************************************
/*> 07_09 ;
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_09
Neutral oscillating dipoles  time averaged force
/$ F(r,v)
= 1/τ1 *∫[dt1,0 to τ1:
1/τ2 *∫[dt2,0 to τ2:
1/2/π*∫[dP1,0 to 2*π:
1/2/π*∫[dP2,0 to 2*π:
1/2/π*∫[∂(φ),0 to 2*π:
1/ π*∫[∂(θ),0 to π: sin(θ)
F(r,O,φ,A1,w1,P1,t1,A2,w2,P2,t2,v) ))))))
/* havent done yet
/* havent done yet
/**********************************************************
/*>
#
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_10
Neutral oscillating dipoles  time averaged force
Simplified  assume two collections of dipoles, spherical symmetry (!?!?!)
/$ F(r,v)
= 1/τ1 *∫[dt1,0 to τ1:
1/τ2 *∫[dt2,0 to τ2:
1/2/π*∫[dP1,0 to 2*π:
1/2/π*∫[dP2,0 to 2*π:
1/ π*∫[∂(θ),0 to π: sin(θ)
F(r,O,φ,A1,w1,P1,t1,A2,w2,P2,t2,v) ))))))
/* havent done yet
/* havent done yet
/********************************************** ;
>>>>>> 7.3  Computation of radial force term ;
/**********************************************************
/*>
p93h0.5 in normal lab experiments that measure gravity,
r2  r1 >> A1, A2, so we will drop A1, A2 terms
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_11
Neutral oscillating dipoles  time averaged radialonly force
/$ F(2  ,1 + ) = q1*q2*r21/r2 + A1  r1^2*ξ
F(2 + ,1 + ) = q1*q2*r21/r2  r1^2*ξ
F(2 + ,1  ) = q1*q2*r21/r2  A1  r1^2*ξ
F(2  ,1  ) = q1*q2*r21/r2  A1 + A2  r1^2*ξ
where
ξ = 1  1/2*(b2  b1)^2  1/2*(b2  b1)^2*cos(θ)^2  1/8*(b2  b1)^4  1/4*(b2  b1)^4*cos(θ)^2 + 3/8*(b2  b1)^4*cos(θ)^4 + ...
/* havent done yet
/* havent done yet
/**********************************************************
/*> 07_12 ;
p93h0.65 for most laboratory measurements of gravity, b2≈b1 & (b2  b1)≈0
For this case only A11 and A22 terms are left where q1=q2=e
is charge of proton, e is charge of electron
?????? CHECK THIS!!
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_12
Neutral oscillating dipoles  radialonly force, time averaged, simplified
/$ F(q2,q1) = q1*q2*r21/r2  r1^2*ξ
/* where
/$ ξ = 1  1/2*(b2  b1)^2  1/2*(b2  b1)^2*cos(θ)^2
 1/8*(b2  b1)^4  1/4*(b2  b1)^4*cos(θ)^2 + 3/8*(b2  b1)^4*cos(θ)^4 + ...
/* havent done yet
/* havent done yet
/**********************************************************
/*>
p94h0.5 sum of "1 terms" should equal 0 > wrong sign for one of the "1"s?
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_13
Neutral oscillating dipoles  radialonly force, time averaged, doublesimplified
/$ F(2 + ,1 + ) = e^2*r21/r2  r1^2*[1]
F(2 + ,1  ) = e^2*r21/r2  r1^2*[1  bpe^2*K1 + bpe^4*k2 ]
F(2  ,1 + ) = e^2*r21/r2  r1^2*[1  bep^2*K1 + bep^4*k2 ]
F(2  ,1  ) = e^2*r21/r2  r1^2*[1  bee^2*K1 + bee^4*k2 ]
/* where
/$ bpp = (b2  b1) for F(2 + ,1 + ) ≈ 0
bpe = (b2  b1) for F(2 + ,1  ) ≈ {A1*w1/c*sin(w1*t1 + P1)}
bep = (b2  b1) for F(2  ,1 + ) ≈ {A2*w2/c*sin(w2*t2 + P2)}
bee = (b2  b1) for F(2  ,1  ) ≈ {A1*w1/c*sin(w1*t1 + P1) + A2*w2/c*sin(w2*t2 + P2)}
K1 = (1 + cos(θ)^2)/2
k2 =  1/8  1/4*cos(θ)^2 + 3/8*cos(θ)^4
/* havent done yet
/* havent done yet
/**********************************************************
/*> 07_14 ;
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_14
Total force between two neutral dipoles
labels : + positive (proton),  negative (electron), dipole1, dipole2
/$ F = F(2 + ,1 + ) + F(2 + ,1  ) + F(2  ,1 + ) + F(2  ,1  )
= e^2*r21/r2  r1^2
*[+ 2*Bpe *Bep *K1
+ 4*Bpe^3*c^2*Bep *k2
+ 4*Bpe *Bep^3*c^2*k2
 6*Bpe^2*c *Bep^2*c *k2
]
/* where
/$ bpp = (b2  b1) for F(2 + ,1 + ) ≈ 0
bpe = (b2  b1) for F(2 + ,1  ) ≈ {A1*w1/c*sin(w1*t1 + P1)}
bep = (b2  b1) for F(2  ,1 + ) ≈ {A2*w2/c*sin(w2*t2 + P2)}
bee = (b2  b1) for F(2  ,1  ) ≈ {A1*w1/c*sin(w1*t1 + P1) + A2*w2/c*sin(w2*t2 + P2)}
K1 = (1 + cos(θ)^2)/2
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas
Component forces between two neutral dipoles, radial
p95h0.0 example  odd powers of sin(wi*ti + Pi) average to zero
/$ 1/τ *∫[∂(t),0 to τ: 1/2/π*∫[∂(φ),0 to 2*π: sin(ω*t + φ)))
= ω/2/π *∫[∂(t),0 to 2*π/ω: 1/2/π*∫[∂(φ),0 to 2*π: sin(ω*t + φ)))
/* set x = w*t
/$ = 1/2/π *∫[∂(t),0 to 2*π: 1/2/π*∫[∂(φ),0 to 2*π: sin(x + φ)))
= 1/2/π *∫[∂(t),0 to 2*π: 1/2/π*∫[∂(φ),0 to 2*π: sin(x)cos(φ)  cos(x)*sin(φ)))
= (1/2/π)^2*∫[∂(t),0 to 2*π: ∫[∂(φ),0 to 2*π: sin(x)cos(φ)  cos(x)*sin(φ)))
= (1/2/π)^2* ∫[∂(φ),0 to 2*π: sin(x)cos(φ)  cos(x)*sin(φ)) )
/* this latter integral (0 to 2*π)
/$ = (1/2/π)^2*(2)*(0  0)
/* 30Aug2015 Howell  I dont like this last step or two, but seems reasonable as result.
However, not just equalpowered sin*cos expressions!
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_15
Component forces between two neutral dipoles, radial
/$ 1/τ*∫[∂(t),0 to τ: 1/2/π*∫[∂(φ),0 to 2*π: sin(ω*t + φ))) = 0
/* havent done yet
30Aug2015 Howell  I dont like this last step or two, but seems reasonable as result.
However, not just equalpowered sin*cos expressions!
/**********************************************************
/*>
Lucas
From (710) and (714)
/$ F_G(r,v)
= 1/τ1 *∫[dt1,0 to τ1:
1/τ2 *∫[dt2,0 to τ2:
1/2/π*∫[dP1,0 to 2*π:
1/2/π*∫[dP2,0 to 2*π:
1/ π*∫[∂(θ),0 to π: sin(θ)
F(r,O,φ,A1,w1,P1,t1,A2,w2,P2,t2,v)
)))))
= w1/2/π*∫[dt1,0 to 2*π/w1:
w2/2/π*∫[dt2,0 to 2*π/w2:
1/2/π*∫[dP1,0 to 2*π:
1/2/π*∫[dP2,0 to 2*π:
1/ π*∫[∂(θ),0 to π: sin(θ)
e^2*r21/r2  r1^2*6*Bpe^2*Bep^2*k2
)))))
= w1/2/π*∫[dt1,0 to 2*π/w1:
w2/2/π*∫[dt2,0 to 2*π/w2:
1/2/π*∫[dP1,0 to 2*π:
1/2/π*∫[dP2,0 to 2*π:
e^2*r21/r2  r1^2*6*Bpe^2*Bep^2 *4/15/π
))))
= w1/2/π*∫[dt1,0 to 2*π/w1:
w2/2/π*∫[dt2,0 to 2*π/w2:
e^2*r21/r2  r1^2*6*Bpe^2*(1/2)^2*4/15/π
))))
= e^2*r21/r2  r1^2*(A1*w1/c)^2*(A2*w2/c)^2*2/5/π
=  2/5/π *e^2*r21/r2  r1^2*(A1*w1/c)^2*(A2*w2/c)^2
/* Note : this is an attractive force ONLY!
where
/$ bpp = (b2  b1) for F(2 + ,1 + ) ≈ 0
bpe = (b2  b1) for F(2 + ,1  ) ≈ {A1*w1/c*sin(w1*t1 + P1)}
bep = (b2  b1) for F(2  ,1 + ) ≈ {A2*w2/c*sin(w2*t2 + P2)}
bee = (b2  b1) for F(2  ,1  ) ≈ {A1*w1/c*sin(w1*t1 + P1) + A2*w2/c*sin(w2*t2 + P2)}
K1 = (1 + cos(θ)^2)/2
k2 =  1/8  1/4*cos(θ)^2 + 3/8*cos(θ)^4
/* in (716) above
/$ sinOdO = ∂(cos(θ)), x=ω*t, dx=ω*∂(t)
/* such that
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_16
Force of gravity from neutral dipoles
Note : this is an attractive force ONLY!
/$ F_G(r,v) =  2/5/π *e^2*r21/r2  r1^2*(A1*w1/c)^2*(A2*w2/c)^2
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas
This looks wrong  improper integral
/$ 1/2/π*∫[∂(φ),0 to 2*π: sin(ω*t + φ))
= 1/2/π*∫[∂(φ),0 to 2*π:
+ sin^2(ω*t)*cos^2(φ)
+ cos^2(ω*t)*sin^2(φ)
+ 2*cos(ω*t)*sin(ω*t)*sinP*cosP
))
= 1/2/π*(π*cos^2(ω*t) + 0 + π*sin^2(ω*t))
= 1/2 *( cos^2(ω*t) + sin^2(ω*t))
= 1/2
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_17
/$ ??? F_G dipoles  ∫sin(ω*t + φ)
1/2/π*∫[∂(φ),0 to 2*π: sin(ω*t + φ)) = 0
/* havent done yet
This looks wrong  improper integral
/**********************************************************
/*>
Lucas
/$ 1/ π*∫[∂(θ),0 to π: sin(θ)*k2)
= 1/ π*∫[∂(θ),0 to π: sin(θ)*(  1/8  1/4*cos(θ)^2 + 3/8*cos(θ)^4))
=  1/ π*∫[∂(cos(θ)),  1 to 1: (  1/8  1/4*cos(θ)^2 + 3/8*cos(θ)^4))
=  1/ π*( cos(θ) , 1 to 1 : ( cos(θ)/8  1/4/3*cos^3(O) + 3/8/5*cos^5(O)))
=  1/ π*( 2/8  1/4/*2/3 + 3/8/*2/5)
= 4/15/π
/* where
k2 =  1/8  1/4*cos^2(O) + 3/8*cos^4(O)
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_18
/$ F_G dipoles  ∫sinO*k2
1/ π*∫[∂(θ),0 to π: sin(θ)*k2) = 4/15/π
/* where
/$ k2 =  1/8  1/4*cos(θ)^2 + 3/8*cos(θ)^4
/* havent done yet
/* havent done yet
/**********************************************************
/*>
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_19
Newtons Universal Law of Gravitation (see 71)
/$ F_G(r) = G*mg1*mg2*r21/r2  r1
F_G(r) = G*mg1*mg2 /r2  r1^2
/* ERROR! should be as I have it
This is WRONG!  missing terms! > eq*q´ etcor e
/**********************************************************
/*>
31Aug2015 Howell
/$ G*mg1*mg2 = 2/5/π*r21*[e*(A1*w1/c)*(A2*w2/c)]^2
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_20
Gravity  equality of?: Newton versus Universal force & neutral dipoles
/$ G*mg1*mg2 = 2/5/π*(A1*w1/c)^2*(A2*w2/c)^2
G*mg1*mg2 = 2/5/π*r21*[e*(A1*w1/c)*(A2*w2/c)]^2
/* WRONG!!
/**********************************************************
/*>
Lucas :=
For two bodies with N1 and N2 atoms of atomic number Z1 & Z2 this becomes :
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_21
F_G for two bodies with N1 and N2 atoms of atomic number Z1 & Z2
/$ G*mg1*mg2 = 2/5/π*N1*Z1*(A1*w1/c)^2*N2*Z2*(A2*w2/c)^2
/* check later again
I DONT LIKE THIS : He has simply stuck in more symbols
This means the original form was incomplete or wrong!!!
/********************************************** ;
>>>>>> 7.4  Corroborating evidence for radiative decay of gravity ;
/**********************************************************
/*>
for simplicity assume N1=N2=1, q1=q2=e, w1=w2=w, m1=m2=m of hydrogen,
A1=A2=A ≤ size of hydrogen atom
/* from wave equation L*f=c, we have w=2*π*f=2*π*c/L
/$ G*m^2 ≥ 2/5/π*e^2*(A*ω/c)^4
= 2/5/π*e^2*(A/c)^4*(2*π*c/L)^4
= 2/5/π*e^2 *(2*π*A/L)^4
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_22
Hydrogen dipole Gm^2 term of gravity
/$ G*m^2 ≥ 2/5/π*e^2*(2*π*A/L)^4
/* check later again
I DONT LIKE THIS : He has simply stuck in more symbols
This could mean the original form was incomplete or wrong!!!
/**********************************************************
/*>
solve for L
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_23
Lambda (L)  wavelength of H2 dipole radiation (gravity)
/$ L^4 ≤ 2/5/π*e^2*16*(π*A)^4/(G*m^2)
/* check later again
Looks OK  going from (722)
/**********************************************************
/*> 07_24 ;
From CRC Handbook of Chemistry & Physics: ;
G := 6.67390e8 ; % cm^3/g/s^2 ;
A_max := 0.37e8 ; % cm ;
e := 4.803e11 ; % g^0.5*cm^1.5/s OR 4.803e11 statC ;
m_e := 1.6726e24 ; % g ;
solve for L
/$ L^4 ≤ 2/5/π*e^2*(2*π*A)^4/G/m_e^2
/* From CRC Handbook of Chemistry & Physics: see constants above
/$ L^4 ≤ 2/5/π*(4.803e11 g^0.5*cm^1.5/s)^2
*(2*π*0.37e8 cm)^4
/ 6.67390e8 cm^3/g/s^2
/ (1.6726e24 ; % g)^2
≤ 1.46 cm or 14.6 mm
/* Howell check
/$ lamda := power (2/5/π*(power e 2)*(power (2*π*A_max) 4)/G/(power m_e 2)) 0.5 ;
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_24
lamda
/$ L^4 ≤ 1.46 cm or 14.6 mm
/* link L^4 ≤ (string power 4 lamda) cm
???OK  same number
/********************************************** ;
>>>>>> 7.6  Computation of nonradial gravitational force term ;
/**********************************************************
/*> 07_25 ;
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_25
( Neutral oscillating dipoles  nonradial force, ??NOT time averaged, doublesimplified
Adapt Lucas07_13 for radialonly force, drop the r21 terms
)
/$ F(2 + ,1 + ) = e^2*(r•β)*r(rβ)/r2  r1^2*[1]
F(2 + ,1  ) = e^2*(r•β)*r(rβ)/r2  r1^2*[1  bpe^2*K1 + bpe^4*k2 ]
F(2  ,1 + ) = e^2*(r•β)*r(rβ)/r2  r1^2*[1  bep^2*K1 + bep^4*k2 ]
F(2  ,1  ) = e^2*(r•β)*r(rβ)/r2  r1^2*[1  bee^2*K1 + bee^4*k2 ]
/* where
/$ bpp = (b2  b1) for F(2 + ,1 + ) ≈ 0
bpe = (b2  b1) for F(2 + ,1  ) ≈ {A1*w1/c*sin(w1*t1 + P1)}
bep = (b2  b1) for F(2  ,1 + ) ≈ {A2*w2/c*sin(w2*t2 + P2)}
bee = (b2  b1) for F(2  ,1  ) ≈ {A1*w1/c*sin(w1*t1 + P1) + A2*w2/c*sin(w2*t2 + P2)}
K1 = (1 + cos(θ)^2)/2
k2 =  1/8  1/4*cos(θ)^2 + 3/8*cos(θ)^4
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas
Neutral oscillating dipoles  nonradial force, time averaged, doublesimplified
Adapt Lucas07_16
/$ F(r,v)
= 1/τ1 *∫[dt1,0 to τ1:
1/τ2 *∫[dt2,0 to τ2:
1/2/π*∫[dP1,0 to 2*π:
1/2/π*∫[dP2,0 to 2*π:
1/ π*∫[∂(θ),0 to π: sin(θ)
F(r,O,φ,A1,w1,P1,t1,A2,w2,P2,t2,v)
)))))
= w1/2/π*∫[dt1,0 to 2*π/w1:
w2/2/π*∫[dt2,0 to 2*π/w2:
1/2/π*∫[dP1,0 to 2*π:
1/2/π*∫[dP2,0 to 2*π:
1/ π*∫[∂(θ),0 to π: sin(θ)
e^2*(r•β)*r(rβ)/r2  r1^2*6*Bpe^2*Bep^2*k2
)))))
= w1/2/π*∫[dt1,0 to 2*π/w1:
w2/2/π*∫[dt2,0 to 2*π/w2:
1/2/π*∫[dP1,0 to 2*π:
1/2/π*∫[dP2,0 to 2*π:
e^2*(r•β)*r(rβ)/r2  r1^2*6*Bpe^2*Bep^2*(3/2/π)
))))
= w1/2/π*∫[dt1,0 to 2*π/w1:
w2/2/π*∫[dt2,0 to 2*π/w2:
e^2*(r•β)*r(rβ)/r2  r1^2
*6*(A1*w1/c)^2*(1/2)*(A2*w2/c)^2*(1/2)
*(3/2/π)
))
=  e^2*(r•β)*r(rβ)/r2  r1^2
*(A1*w1/c)^2*(A2*w2/c)^2
*(9/4/π)
/* ??????Note : is this an attractive force ONLY??? (see 716)
where
/$ bpp = (b2  b1) for F(2 + ,1 + ) ≈ 0
bpe = (b2  b1) for F(2 + ,1  ) ≈ {A1*w1/c*sin(w1*t1 + P1)}
bep = (b2  b1) for F(2  ,1 + ) ≈ {A2*w2/c*sin(w2*t2 + P2)}
bee = (b2  b1) for F(2  ,1  ) ≈ {A1*w1/c*sin(w1*t1 + P1) + A2*w2/c*sin(w2*t2 + P2)}
K1 = (1 + cos(θ)^2)/2
k2 =  1/8  1/4*cos(θ)^2 + 3/8*cos(θ)^4
/* in (712) above (like (716) previously)
/$ sinOdO = ∂(cos(θ)), x=ω*t, dx=ω*∂(t) such that
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_26
Neutral oscillating dipoles  nonradial force, time averaged, doublesimplified
??????Note : is this an attractive force ONLY??? (see 716)
/$ F(r,v) =  e^2*(r•β)*r(rβ)/r2  r1^2(A1*w1/c)^2*(A2*w2/c)^2*(9/4/π)
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas
Adapt Lucas07_18 which is similar
/$ 1/ π*∫[∂(θ),0 to π: sin(θ)*k3)
= 1/ π*∫[∂(θ),0 to π: sin(θ)*(  3/8  9/4*cos(θ)^2 + 15/8*cos(θ)^4))
=  1/ π*∫[∂(cos(θ)),  1 to 1: (  3/8  9/4*cos(θ)^2 + 15/8*cos(θ)^4))
=  1/ π*( cos(θ) , 1 to 1 : ( 3/8*cos(θ)  9/4*2/3*cos^3(O) + 15/8*2/5*cos^5(O)))
=  1/ π*( 3/8*2  9/4/*2/3 + 15/8/*2/5)
= 3/2/π
/* where
/$ k2 =  1/8  1/4*cos(θ)^2 + 3/8*cos(θ)^4
k3 =  3/8  9/4*cos(θ)^2 + 15/8*cos(θ)^4
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_27
Determination of ∫sinO*k3
/$ 1/π*∫[∂(θ),0 to π: sin(θ)*k3) = 3/2/π
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas
Gravity as neutral oscillating dipoles with Universal force
remember  this has been doublesimplified
/$ F_G_total(r,v) = F_G_radial(r,v) + F_G_nonradial(r,v)
/* from (716)
/$ F_G_radial(r,v) = e^2*r21/r2  r1^2*(A1*w1/c)^2*(A2*w2/c)^2*2/5/π
/* from (726)
/$ F_G_nonradial(r,v) = e^2*(r•β)*r(rβ)/r2  r1^2
*(A1*w1/c)^2*(A2*w2/c)^2
*(9/4/π)
))))
/* therefore
/$ F_G_total(r,v)
=  2/5/π*e^2*r21 /r2  r1^2*(A1*w1/c)^2*(A2*w2/c)^2
 9/4/π*e^2*(r•β)*r(rβ)/r2  r1^2*(A1*w1/c)^2*(A2*w2/c)^2
= [ 2/5/π*e^2*r21  9/4/π*e^2*(r•β)*r(rβ)]
/r2  r1^2*(A1*w1/c)^2*(A2*w2/c)^2
/* take 2/5/π*r21*e^2 out of []
/$ = 2/5/π*r21*e^2*(A1*w1/c)^2*(A2*w2/c)^2
/r2  r1^2
*[ 1  45/8/r21*(r•β)*r(rβ)]
/* WRONG expression in Lucas (728) :
/$ F_G_total(r,v) = G*mg1*mg/r2  r1^2*[ r21  45/8*(r21•β)*r21(r21β)]
/* Correct form by substituting with corrected (720) (as per 31Aug2015 Howell) :
/$ G*mg1*mg2 = 2/5/π*r21*[e*(A1*w1/c)*(A2*w2/c)]^2
F_G_total(r,v) = G*mg1*mg/r2  r1^2*[ 1  45/8/r21*(r•β)*r(rβ)]
/* Howells expression Lucas07_27 works OK
(its probably in other papers by Lucas  check later)
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_28
Gravity as neutral oscillating dipoles with Universal force
remember  this has been doublesimplified
/$ F_G_total(r,v) = G*mg1*mg/r2  r1^2*[ r21  45/8*(r21 •β)*r21(r21β)]
F_G_total(r,v) = G*mg1*mg/r2  r1^2*[ 1  45/8/ r21*(r•β)*r (r β)]
/* WRONG expression in Lucas (728) :
Correct form by substituting with corrected (720)
its probably in other papers by Lucas  check later)
/********************************************** ;
>>>>>> 7.8  Origin of Hubbles Law due to gravitational redshifts ;
/**********************************************************
/*> 07_29 ;
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
07_29
Gravitational redshift from star of mass M and radius R
( redshift = GM/R at stellar surface, zero at infinity
G = Newtons universal grav constant
/$ z = ∆L/L = G*M/c^2/R
/*  confirmed experimentally by Puond&Rebka1960
/* havent done yet
/* havent done yet
/********************
>>> Lucas 8  Electrodynamic origin of Inertial forces
/********************************************** ;
>>>>>> 8.1  Introduction ;
/**********************************************************
/*>
Lucas :=
Ratio of gravitational masses of two objects
from Newtons universal gravitational force law
/$ F_g = G*mg1*mg2/r12^2
Fg1 = mg1*g = G*mg1*mE/Re^2
Fg2 = mg2*g = G*mg2*mE/Re^2
/* therefore Fg1/Fg2 = mg1/mg2
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_01
Ratio of gravitational masses of two objects
/$ Fg1/Fg2 = mg1/mg2
/* havent done yet
/* havent done yet  but looks straightforward
/**********************************************************
/*>
Lucas :=
Ratio of inertial masses of two objects
/$ Fi1 = mi1*a1 = mi1*g = mi1
Fi2 = mi2*a2 = mi2*g = mi2
/* therefore Fi1/Fi2 = mi1/mi2
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_02
Ratio of inertial masses of two objects
/$ Fi1/Fi2 = mi1/mi2
/* havent done yet
/* havent done yet  but looks straightforward
/**********************************************************
/*> 08_03 ;
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_03
Surface of Earth  inertial forces are equal to gravitational forces
/$ Fg1/Fg2 = mg1/mg2 = Fi1/Fi2 = mi1/mi2
/* havent done yet
/* havent done yet
/**********************************************************
/*> 08_04 ;
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_04
Newtons 2nd law in accelerating reference frame
/$ sum(F_real) + Fi = mi*a
/* with
/$ Fi = mi*ai
/* havent done yet
/* havent done yet
/**********************************************************
/*> 08_05 ;
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_05
Where mi = mg :
/$ Fi = mi*ai = mg*ai
/* havent done yet
/* havent done yet
/********************************************** ;
>>>>>> 8.2  Derivation of force of inertia from Universal Force Law ;
/**********************************************************
/*>
Lucas
Universal Force Law  first acceleration term as Newtons 2nd law F=ma
Start with (57) Universal_ED_force_with_acceleration
/$ F(r,v,a) =
= q*q´/r^2*
[ + { (1  β^2)*r + 2*r^2/c^2*a } /[r^2  {r(rβ)}^2/r^2]^(1/2)
 (1  β^2)*{ (β•r)*r(rβ) + (r•r)*r(ra/c^2) }
/[r^2  {r(rβ)}^2/r^2]^(3/2)
]
/* Select the acceleration terms only:
/$ F(r,v,a) =
= q*q´/r^2*
[ + 2*r^2/c^2*a /[r^2  {r(rβ)}^2/r^2]^(1/2)
 (1  β^2)*(r•r)*r(ra/c^2) /[r^2  {r(rβ)}^2/r^2]^(3/2)
]
/* from Lucas05_04 [r^2  {r(rb)}^2/r^2] = (1  b^2*sin^2(O))
CHECK this!! ?????????????????
/$ = + q*q´/r*2*a/c^2 /[1  β^2*sin(θ)^2]^(1/2)
 q*q´/r*(1  β^2)*{r(ra/c^2)} /[1  β^2*sin(θ)^2]^(3/2)
= + q*q´/r*2*a/c^2 *[1 + 1/2*β^2*sin(θ)^2]
 q*q´/r*(1  β^2)*{r(ra/c^2)} *[1  3/2*β^2*sin(θ)^2 + 3/2*5/2/2*β^4*sin(θ)^4]
= + q*q´/r*2*a/c^2 *[1 + 1/2*β^2  1/2*β^2*cos(θ)^2]
 q*q´/r *{r(ra/c^2)} *[1 + 1/2*β^2  3/2*β^2*cos(θ)^2 + 3/8*β^4  9/4*β^4*cos(θ)^2 + 15/8*β^4*cos(θ)^4]
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_06
Universal Force Law  first acceleration term as Newtons 2nd law F=ma
/$ F(r,v,a) =
= + q*q´/r*2*a/c^2 *[1 + 1/2*β^2  1/2*β^2*cos(θ)^2]
 q*q´/r *{r(ra/c^2)} *[1 + 1/2*β^2  3/2*β^2*cos(θ)^2 + 3/8*β^4  9/4*β^4*cos(θ)^2 + 15/8*β^4*cos(θ)^4]
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas
Figure 81 is NOT complete in defining the terms! (should define r(2+) etc)
however, it is kind of obvious?
Example : r(2+) is vector (distance) from reference point to charge q(2+)
/$ r(2 + ,1 + ) = r(2 + )  r(1 + ) and A1*f1 = v1
r(2 + ,1  ) = r(2 + )  r(1  )  A1*cos(w1*t + )
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_F8_1
Figure 81 Oscillations of electron in vibrating neutral electric dipole
/$ ( r(2 + ,1 + ) = r(2 + )  r(1 + ) and A1*f1 = v1
r(2 + ,1  ) = r(2 + )  r(1  )  A1*cos(w1*t + )
/* havent done yet
/* havent done yet
/**********************************************************
/*>
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_07
Universal force time&spaceaveraged.over.oscillating neutral dipoles
Force to be compared with Newtons 2nd law F=ma
/$ Fi_neutral_dipoles(r)
= 1/τ1 *∫[∂(t),0 to τ1:
1/ π*∫[∂(θ),0 to π: sin(θ)
1/2/π*∫[∂(φ),0 to 2*π: F(r,O,φ,A1,w1,t)]]]
/* havent done yet
/* havent done yet
/**********************************************************
/*>
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_08
Universal force time&spaceaveraged.over.oscillating neutral dipoles
assume spherical symmetry of neutral dipoles, so integral.over.P=2π
/$ Fi_neutral_dipoles(r)
= 1/τ1 *∫[∂(t),0 to τ1:
1/ π*∫[∂(θ),0 to π: sin(θ)
F(r,O,φ,A1,w1,t)
)))
/* havent done yet
NOTE: key simplification! ignores surface effects locally, but may
be important especially for small clusters at outer edge
/********************************************** ;
>>>>>> 8.3  Derivation of Newtons 2nd law from 1st acceleration term ;
/**********************************************************
/*>
Lucas
/$ Fi(2 + ,1 + ) = e^2*2*a/r2  r1/c^2*[1]
Fi(2 + ,1  ) = e^2*2*a/r2  r1/c^2*[1  bpe^2*K1]
/* where
/$ bpe = (b2  b1) for F(2 + ,1  ) ≈ {A1*w1/c*sin(w1*t1 + P1)}
K1 = (1 + cos(θ)^2)/2
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_09
Universal force time&spaceaveraged.over.oscillating neutral dipoles
Force terms from (86) for 1st acceleration term to order b^4
/$ Fi(2 + ,1 + ) = e^2*2*a/r2  r1/c^2*[1]
Fi(2 + ,1  ) = e^2*2*a/r2  r1/c^2*[1  bpe^2*K1]
/* where
/$ bpe = (b2  b1) for F(2 + ,1  ) ≈ {A1*w1/c*sin(w1*t1 + P1)}
K1 = (1 + cos(θ)^2)/2
/* havent done yet
/* havent done yet  Lucas ??NO square of r2  r1?
/**********************************************************
/*>
Lucas
( Universal force time&spaceaveraged.over.oscillating neutral dipoles
Sum of 1st terms in the [] of the two forces of Lucas08_09 is just 0
/$ F(r,O,φ,A1,w1,t)
= F(2 + ,1 + ) + F(2 + ,1  )
= e^2*2*a/r2  r1/c^2*bpe^2*K1
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_10
Universal force time&spaceaveraged.over.oscillating neutral dipoles
/$ F(r,O,φ,A1,w1,t) = e^2*2*a/r2  r1/c^2*bpe^2*K1
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas
Universal force time&spaceaveraged.over.oscillating neutral dipoles
Using (810) to solve (87)
??Lucas said 88, but used 87, and used P1 not P??
/$ Fi_neutral_dipoles(r)
= 1/τ1 *∫[∂(t),0 to τ1:
1/2/π *∫[∂(φ),0 to 2*π:
1/ π *∫[∂(θ),0 to π: sin(θ)
F(r,O,φ,A1,w1,t)
)))
= 1/τ1 *∫[∂(t),0 to τ1:
1/2/π *∫[∂(φ),0 to 2*π:
1/ π *∫[∂(θ),0 to π: sin(θ)
e^2*2*a/r2  r1/c^2*bpe^2*K1
)))
= 1/τ1 *∫[∂(t),0 to τ1:
1/2/π *∫[∂(φ),0 to 2*π:
e^2*2*a/r2  r1/c^2*bpe^2*(2/3/π)
))
/* substitute
/$ τ1 = 2*π/w1
/* to get :
/$ = w1/2*π*∫[∂(t),0 to 2*π/w1:
e^2*2*a/r2  r1/c^2*(A1*w1/c)^2*(1/2)*(2/3/π)
)
= 2/3/π*e^2*(A1*w1/c)^2/r2  r1*a
= mi1*a
/* But here, Lucas does NOT clearly show that
/$ mi1 = 2/3/π*e^2*2*(A1*w1/c)^2/r2  r1 !!
/* This seems more of a STATEMENT of equivalence,
but he should show that it is reasonable!
/$ (A1*w1/c)^2/r2  r1
/* seems more like an acceleration term?
From (720) :
/$ G*mg1*mg2 = 2/5/π*r21*[e*(A1*w1/c)*(A2*w2/c)]^2
/* so in that case
/$ mg1 ∝ r21^(0.5)*[e*(A1*w1/c)*(A2*w2/c)]
/* (I'm not comfortable about the r21 variable)
where
/$ bpe = (b2  b1) for F(2 + ,1  ) ≈ {A1*w1/c*sin(w1*t1 + P1)}
K1 = (1 + cos(θ)^2)/2
/* *********************
/$ Fi_neutral_dipoles(r)
= 2/3/π*e^2*(A1*w1/c)^2/r2  r1*a
= mi1*a
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_11
Universal force time&spaceaveraged.over.oscillating neutral dipoles
/$ Fi_neutral_dipoles(r)
= 2/3/π*e^2*(A1*w1/c)^2/r2  r1*a
= mi1*a
/* havent done yet
??Lucas said 88, but used 87, and used P1 not P??
/**********************************************************
/*>
Lucas
Used in (811) : This is the wrong integral?!?
/$ 1/2/π *∫[∂(φ),0 to 2*π: sin(ω*t + φ))
= 1/2/π *∫[∂(φ),0 to 2*π:
sin^2(ω*t)*cos^2(φ) + 2*cos(ω*t)*sin(ω*t)*sinP*cosP + cos^2(ω*t)*sin^2(φ))
= 1/2/π *(π*cos^2(ω*t) + 0 + π*sin^2(ω*t)) Lucas dropped the integral notation again!
= 1/2 *( cos^2(ω*t) + 0 + sin^2(ω*t))
= 1/2
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_12
Calculate ∫sin(w*t + P))
/$ 1/2/π *∫[∂(φ),0 to 2*π: sin(ω*t + φ)) = 1/2
/* havent done yet
Used in (811) : This is the wrong integral?!?
/**********************************************************
/*>
Lucas
Machs principle, inertial mass at surface of Earth
with respect to the center of the universe Ruc
from (82) & (721)
Ratio of inertial masses of two objects
/$ Fi1 = mi1*a1 = N1*Z1*(A1*w1/c)^2/Ruc = N1*Z1*(A1*w1)^2/Ruc
Fi2 = mi2*a2 = N2*Z2*(A2*w2/c)^2/Ruc = N2*Z2*(A2*w2)^2/Ruc
/* therefore
/$ Fi1/Fi2 = mi1/mi2
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_13
Machs principle, inertial mass at surface of Earth
with respect to the center of the universe Ruc
/$ Fi1/Fi2 = mi1/mi2
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas
From derivation of F_gravity (716)
/$ F_G(r,v)
=  2/5/π*e^2/r2  r1^2*(N1*Z1*A1*w1/c)^2*(N2*Z2*A2*w2/c)^2
=  G*mg1*mg2/r2  r1^2
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_14
F_gravity from Universal force time&spaceaveraged.over.oscillating neutral dipoles
/$ F_G(r,v) =  G*mg1*mg2/r2  r1^2
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas
From Lucas08_01
from Newtons universal gravitational force law
/$ F_g = G*mg1*mg2/r12^2
/* Ratio of gravitational masses of two objects
/$ Fg1 = mg1*g = G*mg1*mE/Re^2
Fg2 = mg2*g = G*mg2*mE/Re^2
/* therefore
/$ Fg1/Fg2 = mg1/mg2 = (N1*Z1*A1*w1)^2*(N2*Z2*A2*w2)^2 = mi1/mi2
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_15
Ratio of gravitational masses of two objects
/$ Fg1/Fg2 = mg1/mg2 = (N1*Z1*A1*w1)^2*(N2*Z2*A2*w2)^2 = mi1/mi2
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas
815 shows that gravitational and inertial mass at any point in the
universe are equal within a constant k of one another and a radial factor Ruc
/$ mg = k*Ruc*mi
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_16
815 shows that gravitational and inertial mass at any point in the
universe are equal within a constant k of one another and a
radial factor Ruc
/$ mg = k*Ruc*mi
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas
Newtons universal gravitational constant G determine from universal force
Expression below is oversimplified, and Earths proximity may dominate
.over.the average of the spherically symmetric contribution of the rest of the universe
/$ G = 9*π*c^4*Ruc^2/10/e^2
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_17
Newtons universal gravitational constant G determined from universal force
Expression below is oversimplified, and Earths proximity may dominate
.over.the average of the spherically symmetric contribution of the rest
of the universe
/$ G = 9*π*c^4*Ruc^2/10/e^2
/* havent done yet
/* havent done yet
/********************************************** ;
>>>>>> 8.4  Additions to Newtons 2nd law from 2nd acceleration term ;
/**********************************************************
/*>
Lucas
Additions to Newtons 2nd law from 2nd acceleration term.
Lucas07_25 := NOT APPLICABLE?  neutral dipoles, ?nonaccelerating?
Neutral oscillating dipoles  nonradial force, ??NOT time averaged, doublesimplified
Adapt Lucas07_13 for radialonly force, drop the r21 terms
/$ F(2 + ,1 + ) = e^2*(r•β)*r(rβ)/r2  r1^2*[1]
F(2 + ,1  ) = e^2*(r•β)*r(rβ)/r2  r1^2*[1  bpe^2*K1 + bpe^4*k2 ]
F(2  ,1 + ) = e^2*(r•β)*r(rβ)/r2  r1^2*[1  bep^2*K1 + bep^4*k2 ]
F(2  ,1  ) = e^2*(r•β)*r(rβ)/r2  r1^2*[1  bee^2*K1 + bee^4*k2 ]
/* where
/$ bpp = (b2  b1) for F(2 + ,1 + ) ≈ 0
bpe = (b2  b1) for F(2 + ,1  ) ≈ {A1*w1/c*sin(w1*t1 + P1)}
bep = (b2  b1) for F(2  ,1 + ) ≈ {A2*w2/c*sin(w2*t2 + P2)}
bee = (b2  b1) for F(2  ,1  ) ≈ {A1*w1/c*sin(w1*t1 + P1) + A2*w2/c*sin(w2*t2 + P2)}
K1 = (1 + cos(θ)^2)/2
k2 =  1/8  1/4*cos(θ)^2 + 3/8*cos(θ)^4
/* From (86) force terms for 2nd a term to order (v/c)^4 given below.
for velocity terms in [] expressions, consider b2=b1, leaving only A1w1 terms.
Lucas08_06 Universal Force Law  first acceleration term as Newtons 2nd law F=ma.
/$ F(r,v,a) =
= + q*q´/r*2*a/c^2 *[1 + 1/2*β^2  1/2*β^2*cos(θ)^2]
 q*q´/r *{r(ra/c^2)} *[1 + 1/2*β^2  3/2*β^2*cos(θ)^2 + 3/8*β^4  9/4*β^4*cos(θ)^2 + 15/8*β^4*cos(θ)^4]
/* substitute
/$ e^2 = q*q´
/* to get :
/$ = + e^2/r*2*a/c^2 *[1 + 1/2*β^2  1/2*β^2*cos(θ)^2]
 e^2/r *{r(ra/c^2)} *[1 + 1/2*β^2  3/2*β^2*cos(θ)^2 + 3/8*β^4  9/4*β^4*cos(θ)^2 + 15/8*β^4*cos(θ)^4]
/* Screw it  just go with the given formula, look for links to previous equations later.
similar to Lucas07_25 but replace r•b with 1, rb with ra/c^2, add 1/c^2
where
/$ bpp = (b2  b1) for F(2 + ,1 + ) ≈ 0
bpe = (b2  b1) for F(2 + ,1  ) ≈ {A1*w1/c*sin(w1*t1 + P1)}
bep = (b2  b1) for F(2  ,1 + ) ≈ {A2*w2/c*sin(w2*t2 + P2)}
bee = (b2  b1) for F(2  ,1  ) ≈ {A1*w1/c*sin(w1*t1 + P1) + A2*w2/c*sin(w2*t2 + P2)}
K1 = (1 + cos(θ)^2)/2
k5 = (1  3/2*cos(θ)^2)/2
k2 =  1/8  1/4*cos(θ)^2 + 3/8*cos(θ)^4
k3 =  3/8  9/4*cos(θ)^2 + 15/8*cos(θ)^4
k6 = + 3/8  9/4*cos(θ)^2  15/8*cos(θ)^4
/* seems strange  is there a mistake with k6?
***********
/$ F(2 + ,1 + ) = e^2*r(ra/c^2)/c^2/r2  r1*[1]
F(2 + ,1  ) = e^2*r(ra/c^2)/c^2/r2  r1*[1  bpe^2*k5  bpe^4*k6 ]
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_18
Additions to Newtons 2nd law from 2nd acceleration term.
/$ F(2 + ,1 + ) = e^2*r(ra/c^2)/c^2/r2  r1*[1]
F(2 + ,1  ) = e^2*r(ra/c^2)/c^2/r2  r1*[1  bpe^2*k5  bpe^4*k6 ]
/* where
/$ bpp = (b2  b1) for F(2 + ,1 + ) ≈ 0
bpe = (b2  b1) for F(2 + ,1  ) ≈ {A1*w1/c*sin(w1*t1 + P1)}
bep = (b2  b1) for F(2  ,1 + ) ≈ {A2*w2/c*sin(w2*t2 + P2)}
bee = (b2  b1) for F(2  ,1  ) ≈ {A1*w1/c*sin(w1*t1 + P1) + A2*w2/c*sin(w2*t2 + P2)}
K1 = (1 + cos(θ)^2)/2
k5 = (1  3/2*cos(θ)^2)/2
k2 =  1/8  1/4*cos(θ)^2 + 3/8*cos(θ)^4
k3 =  3/8  9/4*cos(θ)^2 + 15/8*cos(θ)^4
k6 = + 3/8  9/4*cos(θ)^2  15/8*cos(θ)^4
/* havent done yet
Lucas07_25 := NOT APPLICABLE?  neutral dipoles, ?nonaccelerating?
seems strange  is there a mistake with k6?
/**********************************************************
/*>
Lucas
Additions to Newtons 2nd law from 2nd acceleration term.
In (818), sum of the first terms in [] of two forces is zero, leaving
/$ F
= F(2 + ,1 + ) + F(2 + ,1  )
= e^2*r(ra/c^2)/c^2/r2  r1*[1]
e^2*r(ra/c^2)/c^2/r2  r1*[1  bpe^2*k5  bpe^4*k6 ]
= e^2*r(ra/c^2)/c^2/r2  r1*[bpe^2*k5  bpe^4*k6]
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_19
Additions to Newtons 2nd law from 2nd acceleration term.
/$ F = e^2*r(ra/c^2)/c^2/r2  r1*[bpe^2*k5  bpe^4*k6]
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas
For integrals of (819), and
/$ (1  3/2*cos(θ)^2)
/* averages to zero.
We use terms up to order b^4 for this term. ?Howell  what?
Similar to : Lucas08_12 :=
/$ 1/2/π *∫[∂(φ),0 to 2*π: sin(ω*t + φ))
= 1/2/π *∫[∂(φ),0 to 2*π:
sin^2(ω*t)*cos^2(φ) + 2*cos(ω*t)*sin(ω*t)*sinP*cosP + cos^2(ω*t)*sin^2(φ))
= 1/2/π *(π*cos^2(ω*t) + 0 + π*sin^2(ω*t))
/* Lucas dropped the integral notation again!
/$ = 1/2 *( cos^2(ω*t) + 0 + sin^2(ω*t))
= 1/2
/* For current integral
/$ 1 /π *∫[∂(θ),0 to π: sin(O)*((1  3*cos(θ)^2/2)))
= 1 /π *∫[∂(cos(θ)),0 to π: ((1  3*cos(θ)^2/2)))
= 1/2/π *(cos  3/3*cos^3(O))(@ 0 to π)
= 1/2/π *( 1  1  (1) + 1)
= 0
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_20
Calculation of averaged ∫sin(w*t + P)
/$ 1/2/π *∫[∂(φ),0 to 2*π: sin(ω*t + φ)) = 0
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas
Additions to Newtons 2nd law from 2nd acceleration term.
/$ F = e^2*r(ra/c^2)/c^2/r2  r1^2*[bpe^2*k5  bpe^4*k6]
/* As with Lucas08_11 :=
Using (819) in (820), instantaneous, nonradial, dipole??
??Lucas said 88, but used 87, and used P1 not P??
/$ Fi_neutral_dipoles(r)
= 1/τ1 *∫[∂(t),0 to τ1:
1/2/π *∫[∂(φ),0 to 2*π:
1/ π *∫[∂(θ),0 to π: sin(θ)*
F(r,O,φ,A1,w1,t)
)))
= 1/τ1 *∫[∂(t),0 to τ1:
1/2/π *∫[∂(φ),0 to 2*π:
1/ π *∫[∂(θ),0 to π: sin(θ)*
e^2*r(ra/c^2)/c^2/r2  r1*bpe^4*k6
)))
= 1/τ1 *∫[∂(t),0 to τ1:
1/2/π *∫[∂(φ),0 to 2*π:
e^2*r(ra/c^2)/c^2/r2  r1*bpe^4*3/2/π
))
/* substitute
/$ τ1 = 2*π/w1
/* to get :
/$ = w1/2*π*∫[∂(t),0 to 2*π/w1:
e^2*r(ra/c^2)/c^2/r2  r1*(A1*w1/c)^4*(3/8)*(3/2/π)
)
= (w1/2*π)*(2*π/w1)*
e^2*r(ra/c^2)/c^2/r2  r1*(A1*w1/c)^4*(3/8)*(3/2/π)
= 9/16/π *e^2*r(ra/c^2)/c^2/r2  r1*(A1*w1/c)^4
= 27/32/π*(A1*w1/c)^4*mi1*r(ra/c^2)
where
bpp = (b2  b1) for F(2 + ,1 + ) ≈ 0
bpe = (b2  b1) for F(2 + ,1  ) ≈ {A1*w1/c*sin(w1*t1 + P1)}
bep = (b2  b1) for F(2  ,1 + ) ≈ {A2*w2/c*sin(w2*t2 + P2)}
bee = (b2  b1) for F(2  ,1  ) ≈ {A1*w1/c*sin(w1*t1 + P1) + A2*w2/c*sin(w2*t2 + P2)}
K1 = (1 + cos(θ)^2)/2
k5 = (1  3/2*cos(θ)^2)/2
k2 =  1/8  1/4*cos(θ)^2 + 3/8*cos(θ)^4
k3 =  3/8  9/4*cos(θ)^2 + 15/8*cos(θ)^4
k6 = + 3/8  9/4*cos(θ)^2  15/8*cos(θ)^4
/* seems strange  is there a mistake with k6?
***************
/$ Fi_neutral_dipoles_nonNewton2nd = 27/32/π*(A1*w1/c)^4*mi1*r(ra/c^2)
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_21
Additions to Newtons 2nd law from 2nd acceleration term.
/$ Fi_neutral_dipoles_nonNewton2nd = 27/32/π*(A1*w1/c)^4*mi1*r(ra/c^2)
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas
/$ 1/ π *∫[∂(θ),0 to π: sin(θ)*k6)
= 1/ π *∫[∂(cos(θ)),  1 to 1: k6)
= 1/ π *k6(from 1 to 1)
/* where
/$ k6 = + 3/8  9/4*cos(θ)^2  15/8*cos(θ)^4
= 3/2/π
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_22
Calculation of
/$ ∫sinO*k
/$ 1/π*∫[∂(θ),0 to π: sin(θ)*k6) = 3/2/π
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas
/$ 1/2/π*∫[∂(φ),0 to 2*π: sin^4(ω*t + φ))
= 1/2/π*∫[∂(φ),0 to 2*π: [sin(ω*t)*cosP + cos(ω*t)*sinP]^4)
= 1/2/π*∫[∂(φ),0 to 2*π:
+ sin^4(ω*t)*cos^4(φ)
+ 4*sin^3(ω*t)*cos^3(φ)*cos (ω*t)*sin (φ)
+ 6*sin^2(ω*t)*cos^2(φ)*cos^2(ω*t)*sin^2(φ)
+ 4*sin (ω*t)*cos (φ)*cos^3(ω*t)*sin^3(φ)
+ cos^4(ω*t)*sin^4(φ)
)
= 1/2/π*[6*π/8*sin^4(ω*t) +6*π/4*sin^2(ω*t)*cos^2(ω*t) + 6*π/8*cos^4(ω*t)]
= 3/8*[sin^2(ω*t) + cos^2(ω*t)]^2
= 3/8
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_23
Calculation of ∫sin^4(w*t + P)
/$ 1/2/π*∫[∂(φ),0 to 2*π: sin^4(ω*t + φ)) = 3/8
/* havent done yet
/* havent done yet
/**********************************************************
/*>
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
08_24
Inertial force law from Universal force
/$ F_I = mi*a  27/32*(A*ω/c)^2*mi*r(ra/c^2)
/* havent done yet
/* havent done yet
/********************
>>> Lucas 9  Structure and harmony of the universe
/********************************************** ;
>>>>>> 9.1  Structure is from symmetry of the Universal Force ;
/**********************************************************
/*>
Lucas09_01 := Lucas05_04 ;
from Lucas05_04 :=
Generalized_potential_U
/$ U(r,v) = q*q´/r*(1  β^2)/(1  β^2*sin(θ)^2)^(1/2)
= q*q´/r*(1  β^2)/[r^2  {r(rβ)/r^2}]^(1/2)
/* where
/$ β=v/c
∂[∂t: U(r,v) =  v•F(r,v,a)
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
09_01
Generalized_potential_U
same as (504)
no need  same as (504)
no need  same as (504)
/**********************************************************
/*>
Lucas Lucas05_07 ;
from Lucas05_07 :=
Universal_ED_force_with_acceleration
/$ F(r,v,a) =
= q*q´/r^2*
[ + { (1  β^2)*r + 2*r^2/c^2*a }
/ [r^2  {r(rβ)}^2/r^2]^(1/2)
 (1  β^2)*
{ (β•r)*r(rβ) + (r•r)*r(ra/c^2) }
/ [r^2  {r(rβ)}^2/r^2]^(3/2)
]
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
09_02
Universal_ED_force_with_acceleration
same as Lucas05_07
no need  same as Lucas05_07
no need  same as Lucas05_07
/********************
>>> Lucas 10  Machs principle and the concept of mass
/********************************************** ;
>>>>>> 10.1  Inertial mass ;
/**********************************************************
/*>
Lucas
from Lucas08_11
/$ Fi_neutral_dipoles(r)
= 2/3/π*e^2*2*(A1*w1/c)^2/r2  r1*a
= mi1*a
/* therefore
/$ mi = 2/3/π*e^2*(A1*w1/c)^2/r2  r1
/* But Lucas summarizes this, for a single vibrating neutral dipole,
consisting of an atomic electron and nuclear proton, to :
/$ mi = 2/3/π*e^2*(A1*w1/c)^2/R/c^2 ???This doesnt make sense???
/* why would ??? :
/$ R*c^2 = r2  r1
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
10_01
Inertial mass of a single dipole of one [proton, electron]
/$ mi = 2/3/π*e^2*(A1*w1/c)^2/R/c^2
/* havent done yet
???This doesnt make sense???
/**********************************************************
/*>
Lucas
for a lump of N atoms each having Z protons and electrons :
NOTE : neutrons count too  perhaps he is using neutron = proton+electron
if so, should reemphasize. R is explain but is problematic!
/$ mi = N*Z*(2/3/π*e^2/R/c^2)*(A1*w1/c)^2
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
10_02
Inertial mass of a lump of of N atoms of some element with Z protons/ electrons
/$ mi = N*Z*(2/3/π*e^2/R/c^2)*(A1*w1/c)^2
/* havent done yet
NOTE : neutrons count too  perhaps he is using neutron = proton+electron
if so, should reemphasize. R is explain but is problematic!
/********************************************** ;
>>>>>> 10.2  Gravitational mass ;
/**********************************************************
/*>
Lucas :=
Factorize radial Fu_G in terms of mi1*mi2*R from (101)
/$ Fu_G
=  (2/5/π)*(e/R)^2*(A1*w1/c)^2*(A2*w2/c)^2*Rh
= G*mg1*mg2/R^2*Rh
=  9/10*π*c^4/e^2*Rh
*{2/3/π/R*e^2/c^2*(A1*w1/c)^2}
*{2/3/π/R*e^2/c^2*(A2*w2/c)^2}
=  9/10*π*c^4/e^2*Rh*mi1*mi2
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
10_03
Factorize radial Fu_G in terms of mi1*mi2*R from (101)
/$ Fu_G =  9/10*π*c^4/e^2*Rh*mi1*mi2
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas
Fug  forces of inertia wrt [center of spiral galaxy mass Mg, center of universe direction R
From (104)
/$ F_I = m*a = m*as*r + m*a0*Rh
/* where
as = accleration with respect to center of galaxy of mass M in direction r
a0 = accleration with respect to center of universe of mass M in direction R
??? does rh > indicate a unit vector? ???
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
10_04
Fug  forces of inertia wrt
Rgh direction to center of spiral galaxy mass Mg
Ruh direction to center of universe mass Mu
/$ F_I = m*a = m*ag*Rgh + m*au*Rh
/* where
ag or as = accleration with respect to center of galaxy
au or a0 = accleration with respect to center of universe
/* havent done yet
/* havent done yet
/**********************************************************
/*> 10_05 ;
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
10_05
Magnitude of the observed acceleration a when ag << au (= as << a0)
/$ a = (ag^2 + au^2)^0.5
= au*(1 + 1/2*ag^2/a0^2 + ...)
= au + 1/2*ag^2/a0 + ...)
/* havent done yet
/* havent done yet
/**********************************************************
/*> 10_06 ;
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
10_06
Force of gravity  wrt center of [universe mass Mu, galaxy mass Mg]
/$ F_G =  G*m*Mu*R/R^2  G*m*Mg*r/r^2
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Lucas
For stability, F_Gravity = F_inertia ?? Maybe not  each could balance???
/$ F_I = m*(au + 1/2*ag^2/au + ...)
= F_G =  G*m*Mu*R/R^2  G*m*Mg*r/r^2
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
10_07
For stability, F_Gravity = F_inertia
/$ F_I = m*(au + 1/2*ag^2/au + ...)
= F_G =  G*m*Mu*R/R^2  G*m*Mg*r/r^2
/* havent done yet
?? Maybe not  each could balance???
/**********************************************************
/*>
rough equivalence of first terms in [F_I, F_G]
/$ m*(1/2*ag^2/au) = G*m*Mg*r/r^2
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
10_08
Approximate, familiar relationship at stability, F_Gravity = F_inertia
/$ m*(1/2*ag^2/au) = G*m*Mg*r/r^2
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Solving for acceleration
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
10_09
acceleration of lump of mass around center of galaxy
at stability, F_Gravity = F_inertia
/$ as = (2*G*Mg*au)^0.5/r
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Use relationship for acceleration in terms of the velocity for circular orbits
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
10_10
acceleration related to circular orbits at stability, F_Gravity = F_inertia
/$ as = vs^2/r = (2*G*Mg*a0)^0.5/r
/* havent done yet
/* havent done yet
/**********************************************************
/*>
Solving for vs
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "no_issue
10_11
speed of circular orbits at stability, F_Gravity = F_inertia
/$ vs = (2*G*Mg*au)^(1/4)
/* havent done yet OK
?OK  constant orbital velocity of planets and moons in a solar system
/*************************************************
>>> 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 "$&" at start of line to help subsequent editing
symTrans_lineTypes are PERSISTENT for multiple lines
 until a symTrans_lineType appears as the first character on a subsequent line.
where mixed (cart [Lucas,HFLN] [symbols,expressions]) are in a line
 put a `^ as the first character of a line to be translated to HFLN [symbols,expressions]
 put a `& as the first character of a line to be translated to Lucas [symbols,expressions]
special codes
#add_eqn  start of summary descriptions and results for a formula from Lucas's book
HFLN  reexpressed equations in HFLN (Howell's FlatLiner Notation)
++
Lists of symbols
[valid, invalid] symbols and their translations are provided in :
"http://www.BillHowell.ca/Projects  big/Lucas  Universal Force/symbol tables/"
24Oct2019 ... I have yet to post the project to my site !?!? ...
tranParn Luca.txt
symList good HFLN bad.txt
symList good HFLN trn.txt
tranExpr Luca.txt
tranParn Luca notes.txt
tranParn HFLN.txt
symList good HFLN.txt
tranExpr HFLN.txt
symList equation numbers.txt
tranParn HFLN notes.txt
symList good Luca.txt
This needs more explanation ion the future ...
++
Prgramming code
A detailed description of the symbols and how they were [checked, translated] is available in a collection of programming files :
"http://www.BillHowell.ca/bash scripts/" :
symbols count in file.sh
symbols compliance POI.sh
symbols compliance special.sh
symbols classify manual.sh
symbols check POI.sh
"http://www.BillHowell.ca/Qnial/MY_NDFS/" :
symbols fix instructions.txt
symbols system loaddefs.ndf
symbols sortarchivemove, postmanualclass.ndf
symbols fix.ndf
symbols count in file.sh.txt
symbols log.txt
symbols extract.ndf
symbols translate.ndf
symbols compliance special.sh.txt
symbols check POI.sh.txt
symbols classify manual.sh.txt
symbols compliance POI.sh.txt
symbols zProcess notes.txt
/*_endCmd
/********************************************
>>>>>>>>> HFLN = Howell's FlatLiner Notation !!!!!!!!!!!!!! 31May2016
/*/*$ cat >>"$p_augmented" "$d_Lucas""context/Howells flatline notation short description.txt"
All material herein follows "Howell's 'FlatLine' formatting conventions" as specified in my document :
"$d_Lucas""document/math Variables, notations, styles YYMMDD HHhMMm.txt"
This is important, as my notations helped me to be more clear on the concepts and contexts being used. This was a lot of work and headaches, but it did result in a greater ease of writing the derivations, and often the notation made conceptual errors much more visible and trackable.
An obvious DISADVANTAGE of my notations is that it will be unfamiliar to others, and considerable time would be required for a reader to become functionally familiar with the notation. . However, it is (theoretically) relatively easy (if practically long and tedious) to program a conversion of the final results of each subsubsection from my notation to the standards used by Bill Lucas and the physics community, for example with a preprocessor to Tex or something like that.
Rather than reediting this entire file to use my "most uptodate" notations, I have simply added the HFLN notations to my comments for SOME of the expressions. Problems have not been fully resolved!!
WARNING : Because I rapidly added the expressions for HFLN, many mistakes remain.
Being precise about the context for each variable has been a persistent problem.
/*_endCmd
/**********************
>>>>>> Document build short description
/*/*$ 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 "/*/*$", denotes deactivated commands.
This is done automatically for each pass
As of 23Oct2019, you must manually rerun "$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 searchreplace, such as :
search : /*$(.*)n
replace : /*$1n/*_endCmdn
Hopefully, if you are not using kwrite, you will have equivalent regular expression searhreplace (if not change text processors!!). Emacs would have it, probably vi as well, although I don't use them.
++
Table of Contents, list of equations
The script "$d_bin""txtDoc insert indexes.sh" inserts the :
/*_Insert_Table_of_Contents
202: Lucas's Dedication
230: Introduction
312: Lucas 4  Derivation of the Universal Electrodynamic Force Law for constant velocity
316: 4.1  Proper Axioms of Electrodynamics
342: Fundamental Equations Of Electrodynamics ;
623: Howell  Appendix A, Derivation of the BiotSavart and Grassman form of Amperes Law
637: 4.2  Derivation of Electrodynamic Force Law
2304: Summary  Derivation of the relativistic correction factor (1  β^2*sin(Aθpc(POIo(t),t=0))^2)^(3/2)
2351: Highly restrictive conditions for dropping the (cos  1) terms /%
2372: Targeted results /%
2391: Multiple conflicting hypothesis /%
2404: Does the relativistic correction factor matter? /%
2506:
2778: Bottomup (2b1)
4192: Lucas 5  Extension of the Universal Force Law to include acceleration
4198: 5.1  Generalized electromagnetic potential U(r,v)
4445: 5.2  Acceleration fields and radiation ;
4647: Lucas 6  Extension of the Universal Force Law to include radiation reaction da/dt
4651: 6.3  Derivation of nonrelativistic radiation reaction force ;
4768: 6.4  Derivation of relativistic radiation reaction force ;
4832: Lucas 7  Electrodynamic origin of gravitational forces
4836: 7.1  Introduction, Electrodynamic origin of gravitational forces ;
4980: 7.2  Origin of gravitational forces ;
5106: 7.3  Computation of radial force term ;
5431: 7.4  Corroborating evidence for radiative decay of gravity ;
5509: 7.6  Computation of nonradial gravitational force term ;
5677: 7.8  Origin of Hubbles Law due to gravitational redshifts ;
5700: Lucas 8  Electrodynamic origin of Inertial forces
5703: 8.1  Introduction ;
5808: 8.2  Derivation of force of inertia from Universal Force Law ;
5921: 8.3  Derivation of Newtons 2nd law from 1st acceleration term ;
6180: 8.4  Additions to Newtons 2nd law from 2nd acceleration term ;
6449: Lucas 9  Structure and harmony of the universe
6452: 9.1  Structure is from symmetry of the Universal Force ;
6506: Lucas 10  Machs principle and the concept of mass
6509: 10.1  Inertial mass ;
6564: 10.2  Gravitational mass ;
6738: APPENDICES
6741: Future extensions of the Universal Force
6756: Gaussian versus SI units ;
6770: Symbol checking and translation  short description
6856: HFLN = Howell's FlatLiner Notation !!!!!!!!!!!!!! 31May2016
6876: Document build short description
6961: REFERENCES
/*_Insert_equations
352: (401) Generalized_Amperes_Law
403: (402) Faradays_Law
445: (403) Gauss_Electrostatic_Law
479: (404) Gauss_Magnetostatic_Law
508: (405) Lenz_Induction_Law
528: (405a) E as sum of E0 & Ei
545: (406) Lorentz_Force_Law
595: (407) Galilean_transformation
645: (408) Induced_magnetic_flux_density from Amperes law
683: (409) Frame_transformation_info_lost by Maxwell
726: (410) Galilean_transformation
740: (411) E&B_fields_static_plus_induced
762: (412) E Galilean transformation particle to observer frames
777: (413) Total B magnetic flux density as induced from E0 + Ei
842: (414) B&E point charge  substituted Amperes law
918: (414a) Point particle and symmetry
942: (415) E,B for symmetry point charge @v_const
1040: (416) E,B for symmetry point charge @v_const
1138: (417) Spherical coordinate transforms
1177: (418) Changing magnetic flux linked by a circuit proportional to induced E field around the circuit
1211: (419) E,B for symmetry point charge @v_const  Stokes theorem
1297: (420) Convective_derivative
1321: (421) convective derivative of Total magnetic flux density Bi
1443: (422) KelvinStokes integration of convective derivative of Bi total
1474: (423) Faradays_Law_for_rest_circuit integral form E,B
1549: (424) E&B for [Faradays + part/obs frameTrans]  towards FU_Faradays_Law
1631: (425) E&B for [Faradays + part/obs frameTrans]  towards FU_Faradays_Law
1665: q,E,Bi)
1709: (427) Lorentz Force
1891: (428a) Faradays_law_spherical_coords  ∇´Ei(ro  vo*t,t) term
1968: Bi(ro  vo*t,t)] term
2003: (429a) Faradays_law_spherical_coords  1st term
2088: (429b) Faradays_law_spherical_coords  2nd term
2110: (430) Faradays_law integrated over θ
2256: (431) From Lenzs law and symmetry of local forces
2286: (431a) Machs principle  Lenz works, SRT & covariant Maxwell fail
2414: (432) EIods(POIo,t=0,1st stage), F therefore E balance  iteration #1 on (430)
2591: (433) EIods(POIo,t=0, 2nd stage), F therefore E balance
2719: (434) EIods(POIo,t=0, 2nd stage), K_2nd from taking partial derivatives wrt time
2889: (435) E0ods(POIo,t) truncated expression with ONLY E0ods(POIo,t) terms
3050: (436) ETods(POIo,t) expression with ONLY ETods(POIo,t) terms
3085: (437) Er and the binomial series, leading to the relativistic correction factor
3668: (438) Binomial_expansion_for_E0_terms
3711: (439) E(r,v) for constant velocity, nonpoint charge, observer reference frame
3775: (440) Gauss_Electrostatic_Law
3793: (441) L(v) expression for Gauss law for electric charge
3893: (442) Special integral with binomial series (1  b^2*sin^2(O))^(3/2)
3972: (4_43) E&B_fields_self_consistent
4022: (444) F_total by moving charge distribution on a test charge q'
4125: (445) Vector identities for Lorentz Force derivation
4150: (446) Vector_operations used for the Lorentz force
4202: 05_01 ;
4235: 05_03 ;
4380: 05_08 ;
4449: 05_11 ;
4468: 05_12 ;
4487: 05_13 ;
4840: 07_01 ;
4861: 07_02 ;
4903: 07_04 ;
5015: 07_08 ;
5057: 07_09 ;
5133: 07_12 ;
5192: 07_14 ;
5475: 07_24 ;
5513: 07_25 ;
5681: 07_29 ;
5753: 08_03 ;
5772: 08_04 ;
5792: 08_05 ;
6621: 10_05 ;
6638: 10_06 ;
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 "/*$" command approach. Instead, one might want to view a file during the process of reading, or the file is an [image, spreadsheet] which cannot be directly incorporated in a txtDoc. Therefore, a bash line is provided, which can be copypasted into a terminal window to show the file. I have yet to provide default window [size, position] coding, which will be applicationspecific.
Note that a simple "comment" symbol "/*" is used in these cases.
 view a text file, in this case using my preferred text editor :
/* kwrite ""
 more general form, where readers can predefine an environmental variable in lieu of using a specific program :
/*$ eval '$txtEditor ""'
 view an image file, again using a specific image viewer :
/* eog ""
 more general form, where readers can predefine an environmental variable in lieu of using a specific program :
/*$ 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 0963447157
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 ISBN13: 9781482328943
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 9780471309321
Erwin Kreyszig 1972 "Advanced Engineering Mathematics, Third Edition" John Wiley & sons, ISBN 0471507288
COOL!
https://en.wikipedia.org/wiki/Oliver_Heaviside
"... Oliver Heaviside FRS[1] (/ˈɒlɪvər ˈhɛvisaɪd/; 18 May 1850 – 3 February 1925) was a selftaught 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 coformulated 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