www.BillHowell.ca's Background math for a review of Bill Lucas's book the "Universal Force, Volume I" /*$ echo "version= $date_ymdhm" >>"$p_augmented" This file is : /*$ echo "$p_augmented" >>"$p_augmented" /********************** >>> SUMMARY I've no doubt that results here are available in basic physics textbooks, but : it was important for me to work the material in detail in order to better understand the origins of key expressions this document provides a broad, detailed] basis to help users become familiar with "Howell's flatliner notation" equations parametric in t have been developed, both as a check on errors in my derivations, and to provide a the content is easily [edititable, auditable, extendible] once "Howell's flatliner notation" is understood. (see below) I've made several checks on taking a "cheating" apporoach to derivatives (for example, taking derivatives of scalar functions rather than their vector origins). Usually the cheating approach works fine. The level of detail provided allows the reader a clear view of my own errors and shortcomings. /*$ cat >>"$p_augmented" "$d_Lucas""context/summary - general.txt" /*_endCmd /*$ cat >>"$p_augmented" "$d_Lucas""context/text editor - how to set up.txt" /*_endCmd *********************** TABLE OF CONTENTS /*_Insert_Table_of_Contents EQUATIONS : /*_Insert_equations For instructions on how to update the Table of Contents and Equations, see the section "Document build short description" at the end of this document. There is currently a problem of both lists above "shifting" the line number counts. /********************************************** waiver, copyright /*$ cat >>"$p_augmented" "$d_Lucas""context/waiver, copyright.txt" /*_endCmd /********************************************** >>> Lucas's Dedication (This is copied directly from his book.) /*$ cat >>"$p_augmented" "$d_Lucas""context/Lucas dedication.txt" /*_endCmd /********************************************** >>> Introduction /*$ cat >>"$p_augmented" "$d_Lucas""context/introduction.txt" /*_endCmd /********************************************** >>> I. Basics /***************************************** >>>>>> [Observer, particle, ether] reference frames /*$ cat >>"$p_augmented" "$d_Lucas""context/reference frames.txt" /*_endCmd /***************************************** >>>>>> Formulations of electrodynamics It is important for me to emphasize several key [equations , systems of models] that will be well known to experts in the area, for those like me who tend to forget. /********************* >>>>>>>>> Maxwell's equations (... I need to explain the differences between the classic 4-vactor Maxwell equations, and those of Lucas" ...) perhaps provide a table... 31Mar2016 Big questions at the base : Normally, Maxwell's equations relate RATE OF CHANGE of B to E : as with (4-2) Faraday'sd Law, but not like (4-1) Generalized Ampere's Law (actulally latter probably OK?) Lenz's Law - is KEY, need proof from other approaches as this is a HUGE simplification! Barnes iterations - how does this look fundamentally-geometrically? /********************* >>>>>>>>> Covariant version Jackson 1999 p??h?? Equation ?? /********************* >>>>>>>>> ?Heaviside? 4-vector formulation /********************* >>>>>>>>> ?Hamilton's? quaternion formulation Can one consider this to be the proper basis? /********************* >>>>>>>>> Lucas's equivalent /********************* >>>>>>>>> Ed Dowdye Jr's "Extinction shift principle" /***************************************** >>>>>> Questions ******************** >>>>>>>>> Random, scattered questions 1. Is the particle frame of reference adequate for a "point of interest" (observer or particle reference frame) when charges have finite size, or is a "Point of Interest" reference frame required to make calculations tractable? 2. With distributed charges, even within the particle frame of reference calculations become much more challenging. 3. Do Lucas's formulae properly integrate the contributions of distributed charge? He assumes toroidal loops, but the calculations seem to show only one of infinitely many possible arranagments of two toroidal loops, and don't show the effect of neutral dipols within U atoms, or molecular compounds of much greater size and complexity. 4. After a lifetime of being "programmed" to believe in a Dirac Nucleus and electron shell, is there a better way to transition to Lucas's point of view to better understand it? 5. I am guessing somewhat at several details of the separate [particle, observer] frames of reference. /********************* >>>>>>>>> Initial linearity assumptions, but non-linear models From Ampere, Faraday, Oersted, etc - basics "Laws" are linear. But the Universal Force (and Jeffimenko's causality, probably Randall Mills) is non-linear, meaning that the derivations are self-inconsistent? /********************* >>>>>>>>> Superluminal speeds IMPORTANT NOTE : There is an explicit belief in mainstream physics, AND IN LUCAS'S book, an implicit assumption, that speeds can never exceed the speed of light in a vacuum, c. However, that should NOT be a limitation of Lucas's Universal force!! There is therefore no guarantee that v/c is <1, and therefore that : Nyet : 0 <= [(1 - lambda(v)), (1 - beta^2) <= 1 This affects derivatives! /********************* >>>>>>>>> Howell's use of the Kahan formulation for a "Scalar derivative of the norm of a vector function" See below : "Scalar derivative of the norm of a vector function" : In working with vector derivatives, I have blindly made a key assumption regarding the Kahan formulation : d||z|| = u_T dotPRod dz / ||z|| where u_T is the linear functional dual to z wrt ||...|| But is this correct? Or, more specifically, under what conditions will I run into trouble with this assumption? Given the importance of vector derivatives, I really need to look into this much more closely! Likewise, the same question applies to my use of "Scalar integral of the norm of a vector function" : Therefore, the integral is : |x| = ∫[dx : x/|x| } for x≠0. /********************* >>>>>>>>> Time delays / Field lag - Temporal equivalence within a frame of reference for "short" distances? For a particle moving through space at constant velocity with no interactions (eg [E,B, other forces) along the way, we will assume that the [E,B] field structures originating from the particle do not change, with an important limitation. If one assumes that the fields extend instantaneously to the ends of the universe, then there is no need to consider "field lag". This seems to be Lucas's assumption. However, assuming a speed of propagation of electromagnetic fields to equal that of electromagnetic radiation then one must consider field lag. For the purposes of Chapter 4, we'll assume that rps_max is limited in the analysis to a scale within which lag effects are negligible, which also means that the particle motion has been in effect long enough BEFORE we start the analysis to establish the "field lag" to the extent of rs. /********************* >>>>>>>>> How do toroidal [electrons, protons] behave with [spin,rotations, accelerations]? Given the toroidal ring model favoured by Lucas and his sources [?Bergman, ???], what happens when a single [electron, proton, atom, molecule, clumped of matter] are subject to non-rectilinear accelerations? My first guess is that gyroscopic forces of some sort would be involved, and would tend to cancel one another in a clump of solid matter consisting of large numbers of molecules in random orientations. Neutral atomes might have the same property, with gyroscopic forces balancing out? Might this relate to [polarization, spin, color] in subatomic physics? /********************* >>>>>>>>> Do toroidal [electrons, protons] [spin, oscillate] via [precession, obliquity]? For ?names - Bergman, etc??? toroidal [electron, proton] structural theory favoured by Lucas, do gyroscopic-like restraints allow for free spin of the atom? Presumably at least oscillation occurs? Note that recently, Lucas has claimed that recent NIST data shows that neutrons are NOT a different sub-atomic particle, and are merely a tightly bound electron-proton pair. This is reminiscent of the beta particle (helium nucleus). /********************* >>>>>>>>> PROBLEM - When is an induce field "real"? see the 6 equations in Lucas p64 This is a question I have from equations [(4-13),(4-14),(4-15)] /* Generalized Ampere's Law -> "v" will be very different for different reference frames. - If I assume that the Bi field is the same in all reference frames, then v is a velocity relative to WHAT? - Presumably it must be relative to the E0 field? From this, I conclude that Equation 1 - implies that the B field DOES depend on the observer frame! So if : - you moved with a point charge, you would see NO B induced field? - a very high-speed passing observer would see arbitrarily large B fields? - many observers moving past the point at different relative velocities will see different B fields Weird, I didn't think of these things. Reminds me that a constant current in a wire has a B field, but no E field (which I hadn't thought of)! /********************* >>>>>>>>> Is my application of Lenz's Law legitimate? /***************************************** >>>>>> Major discrepancies between my own derivations and those of Lucas /********************* >>>>>>>>> Form of expression for "∂[∂(t): BTpdv(POIo(t),t)] = ∂[∂(t): BTodv(POIo,t)]" /*$ cat >>"$p_augmented" "$d_augment""d-dt BTpdv vs BTodv discrepency with Lucas.txt" /*_endCmd /********************* >>>>>>>>> Lucas's vt*[cos(Aθpc(POIo(t),t)) - 1] term see Lucas p71h0.25 I have consistently been unable to come up with this term!!! See Equation (3) in the previous sub-sub-section for my current expression. /********************* >>>>>>>>> B X v field is not electrostatic in nature I do not yet have a good grasp of this. My current understanding is that while ETodv(POIo,t) DOES use an iterative process, this process does NOT feed back into "B X v" as Lucas's point (from Cullick[8], Hooper[10, and Spencer[12]) is that E does NOT arise from the induced "B X v" E fields. At present - this statement seems contradictory, as Lucas does use the TOTAL ETodv(POIo,t) to calculate the total BTodv(POIo,t) field? /********************************************** >>> II. Derivations for a POIp = POIo(t) fixed in the particle reference frame (RFp) /***************************************** >>>>>> Basic measures /********************* >>>>>>>>> Figure "Basic measures for for the particle reference frame RFp, using POIp=POIo(tx)" Run command to see $ eog "$d_images""Howell - Chapter 4 - POIo basic - cropped.png" & Run command to see $ eog "$d_images""Howell - Chapter 4 - dp[dt _ Rpcs(POIo,t)] - cropped.png" & Reminder : The [particle, observer] reference frames [(RFp),(Rfo)] have the same scaling and orientation, and at time t=0 their origins coincide, being an exact match at that time apart for the motion of (RFp) with the particle. Reminders for Chapter 4 : Vonv(PART) = constant [Particle, observer] frames of reference (RFp) & (RFo) are IDENTICAL [scale, rotation, etc] at time t=0, apart from the given that the particle's reference frame (RFp) moves with velocity v relative to RFo. Reminders for (POIp) - Point Of Interest that is FIXED in the particle reference frame (RFp) : [Rpc,Opc,Ppc] and their derived concepts, are NOT functions of time, i.e. (POIp). Their derivatives with respect to time are zero. [Roc,Ooc,Poc], their derived concepts, and their derivatives are all functions of time, i.e. (POIp(t),t) OBJECTIVES : Where convenient, express variables as functions of FIXED variables : (POIp) -> -Vodv(PART),t,Rpcv(POIp),Aθpc(POIp),Aφpc(POIp) (POIo) -> Vodv(PART),t,Rocv(POIo),Aθoc(POIo),Aφoc(POIo) /********************* >>>>>>>>> Rpcv(POIp), Aθpc(POIp), Aφpc(POIp) are constants As per the intRoduction to this section, : This section deals with (POIp) - Points Of Interest(POI) which are fixed with respect to the particle reference frame (RFp), and that move with constant relative velocity Vonv(PART) with repect to the observer. (mathH) Rpcv(POIp) = constant (endMath) (mathH) Aθpc(POIp) = constant (endMath) (mathH) Aφpc(POIp) = Aφoc(POIp(t),t) = constant (endMath) /********************* >>>>>>>>> Rpcv(POIo(t),t) /*$ cat >>"$p_augmented" "$d_augment""Rpcv.txt" /*_endCmd /********************* >>>>>>>>> Rpcs(POIo(t),t) /*$ cat >>"$p_augmented" "$d_augment""Rpcs.txt" /*_endCmd /********************* >>>>>>>>> sin(Aθpc(POIo(t),t)) /*+-----+ (RFp) basis From "R_OPI2_ocs(POIo) = RθPI2pcs(POIo)" : /% (1)* RθPI2pcs(POIo) = Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t)) = constant Therefore : (mathH) sin(Aθpc(POIo(t),t)) = RθPI2pcs(POIo)/Rpcs(POIo(t),t) (endMath) /*+-----+ (RFo) basis From "Rocv(POIo), Aθoc(POIo), Aφoc(POIo) are constants" : /% (mathH) Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) (endMath) So : sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) / Rpcs(POIo(t),t) = Rocs(POIo)*sin(Aθoc(POIo)) / { [Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ]^2 + (Rocs(POIo)*sin(Aθoc(POIo)))^2 }^(1/2) Summarizing : (mathH) sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) / {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) (endMath) /*+--+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (dimensionless). /********************* >>>>>>>>> cos(Aθpc(POIo(t),t)) From (4)&(2) : /% cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / Rpcs(POIo(t),t) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { [Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ]^2 + (Rocs(POIo)*sin(Aθoc(POIo)))^2 }^(1/2) Summarizing : (mathH) cos(Aθpc(POIo(t),t)) = [Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] / {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2}^(1/2) (endMath) /*+-----+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (dimensionless). /*sin^2 + cos^2 = 1 /% (5) sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) (6) cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) /*OK by inspection. /********************* >>>>>>>>> R_O0_pcs(POIo(t),t) See Figure "Basic measures for a POIo" : /*+-----+ (RFp) basis /% (mathH) Rθ0pcs(POIo(t),t) = Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) (endMath) /*LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (length). /*+-----+ (RFo) basis Again using Figure "Basic measures for a POIo" : Distance of R_O0_pcs(POIo) from the (RFp) origin in O0ch direction (i.e. along L(PART)). Note that the notation "_O0_" is a mnemonic for theta = O (capital O) = 0 (zero) radians : /% Rθ0pcs(POIo) = Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) = Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t (mathH) Rθ0pcs(POIo(t),t) = Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t (endMath) /********************* >>>>>>>>> RθPI2pcs(POIo(t),t) Distance of RθPI2pcs(POIp) = R_OPI2_ocs(POIo) from L(PART) in Ppch=Poch direction (i.e. perpendicular to L(PART)). Note that the notation "_OPI2_" is a mnemonic for theta = O (capital O) = PI/2 radians. Note that this measure is a constant for any POIo or (POIp(t),t), independent of the motion of the particle, and is therefore a convenient basis for (RFp) claculations for (POIo) (Points Of Interest fixed in the particle frame of reference). Note : as this sub-sub-sub-section was moved, original augmented equation numbering is used here (to avoid having to edit all references elsewhere in the document). This should ultimately be cleaned up throughout the document. /*+-----+ (RFp) basis /% (mathH) RθPI2pcs(POIo(t),t) = Rpcs(POIp)*sin(Aθpc(POIp)) = constant (endMath) /*+-----+ (RFo) basis From (3p) : /% 3p) RθPI2pcs(POIo(t),t) = Rpcs(POIo(t),t) *sin(Aθpc(POIo(t),t)) = Rocs(POIo) *sin(Aθoc(POIo)) ...[Rocv(POIo),Aθoc,Aφoc] don't change with t (mathH) RθPI2pcs(POIo(t),t) = Rocs(POIo)*sin(Aθoc(POIo)) = constant (endMath) /*+--+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (length). /********************* >>>>>>>>> K0, K1, K2 for use in differentiations /% (mathH)/* for use in differentiations /% K0 = 3/2*β^2*Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2 (endMath) (mathH)/* for use in differentiations /% K2 = (-1)*λ(Vons(PART))*Q(PART)*Rpcs(POIo(t),t)^(-2) (endMath) (mathH)/* not used in derivations, see "???" /% K1 = (-3)*β^2*Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t)^(-5)*Vons(PART)*t*(cos(Aθpc(POIo(t),t)) - 1) (endMath) /* where /% (mathH) f_sphereCapSurf(x) = β*Rocs(POIo)^2*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))*∂[∂(t): x]] (endMath) /********************* >>>>>>>>> K0(t=0), K1(t=0), K2(t=0) in terms of E0ods(POIo,t) for relativistic factor, when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! substitute E0ods(POIo,t) = Q(PART)/Rpcs(POIo(t),t)^2 given that : t=0 particle and reference frames are the same, x-axis is trajectory of particle, in direction of motion wrt RFo POIo is on the trajectory of the particle in the direction of Vons(PART) then (mathH) Rocs(POIo) = Rpcs(POIo(t),t=0) when : t=0, RFp=RFo @t=0, POIo is on the trajectory of the particle in the direction of Vons(PART) (endMath) <--???--| /% (mathH)/* when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! /% K0(t=0) = E0ods(POIo,t=0)*3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 (endMath) (mathH)/* when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! /% K2(t=0) = -E0ods(POIo,t=0)*λ(Vons(PART)) (endMath) (mathH)/* not used in derivations, see "???" /% K1(t=0) = 0 (endMath) /* derivation of K1(t=0) /% K1(t=0) = -3 *β^2*Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(-5)*Vons(PART)*t*(cos(Aθpc(POIo(t),t=0)) - 1) = -3 *β^2*Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(-5)*Vons(PART)*0*(cos(Aθpc(POIo(t),t=0)) - 1) = 0 /* IFFY /********************* >>>>>>>>> EIods(POIo,t=0,ith stage) /* Note that this is a clean definition of a not-actually-recursive process : 05Oct2019 still need to do derivations, proofs must link to lines in "Howell - math of Lucas Universal Force.txt" /% (mathH) EIods(POIo,t=0,1st stage) = K_1st + f_sphereCapSurf(EIods(POIo,t)) (endMath) (mathH) EIods(POIo,t=0,2nd stage) = K_1st + f_sphereCapSurf(EIods(POIo,t=0,1st stage) (endMath) (mathH) EIods(POIo,t=0,ith stage) = K_1st + f_sphereCapSurf(EIods(POIo,t=0,(i-1) stage)))} (endMath) /********************* >>>>>>>>> K_1st, K_2nd for use in differentiations (mathH) K_1st = K0 + K2 (endMath) (mathH)/* differentiable form /% K_1st = + Q(PART) *( 3/2 *β^2*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2 - λ(Vons(PART)) *Rpcs(POIo(t),t)^(-2) ) (endMath) /* using /% 2940:(mathL)/* HIGHLY restricted! at time t=0 This means that the [observer, particle] reference frames are exactly the same at t=0. /% Rocs(POIo) = Rpcs(POIo(t),t=0) K_1st = + 3/2*β^2*Q(PART)*Rpcs(POIo(t),t=0)^(-2)*sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART)) *Q(PART)*Rpcs(POIo(t),t=0)^(-2) = Q(PART)*Rpcs(POIo(t),t=0)^(-2) *( + 3/2 *β^2 *sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART)) *1 ) /* using /% 1205:(mathH) E0ods(POIo,t) = E0pds(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2 K_1st = E0ods(POIo,t=0) *( + 3/2 *β^2 *sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART)) *1 ) (mathH)/* HIGHLY restrictive conditions : t=0, RFo(t=0) = Rfp(t=0), Rocs(POIo) = Rpcs(POIo(t),t=0) This means that the [observer, particle] reference frames are exactly the same at t=0 (other than motion). drop as roundoff error : f_sphereCapSurf expression see "textIn - relativistic factor, intermediate symbols, restrictive conditions.txt" /% K_1st = E0pds(POIp)*3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 - E0pds(POIp)*λ(Vons(PART)) (endMath) K_2nd = β*Rocs(POIo)^2 *{ + 21/2 *β^3*Q(PART)*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-6)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t)) ] - 2 *λ(Vons(PART)) *β *Q(PART)/Rocs(POIo) *Rpcs(POIo(t),t)^(-3)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t)) *cos(Aθpc(POIo(t),t)) ] } /* using /% 2715:(mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^1] = sin(Aθpc(POIo(t),t=0))^2/2 2717:(mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^3] = sin(Aθpc(POIo(t),t=0))^4/4 K_2nd = β*Rocs(POIo)^2 *{ + 21/2 *β^3*Q(PART)*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4/4 - 2 *λ(Vons(PART)) *β *Q(PART)/Rocs(POIo) *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t=0))^2/2 } = + 21/2/4 *β^4*Q(PART) *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4 - 2/2 *λ(Vons(PART)) *β^2*Q(PART)*Rocs(POIo) *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t=0))^2 = + 21/8 *β^4*Q(PART) *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4 - λ(Vons(PART)) *1 *β^2*Q(PART)*Rocs(POIo) *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t=0))^2 (mathH) K_2nd = + 21/8 *β^4*Q(PART) *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4 - λ(Vons(PART)) *1 *β^2*Q(PART)*Rocs(POIo) *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t=0))^2 (endMath) /********************* >>>>>>>>> K_1st(t=0), K_2nd(t=0), K_3rd(t=0) in terms of E0ods(POIo,t) for relativistic factor, when [t=0, RFp=RFo @t=0], use only AFTER differentiations!!! /% ????:(mathH) K_1st = 3/2*β^2*Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t=0)^(-5)*sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART))*Q(PART)*Rpcs(POIo(t),t=0)^(-2) for use in differentiations 1049:(mathH) Rocs(POIo) = Rpcs(POIo(t),t=0) when : t=0, RFp=RFo @t=0 K_1st = 3/2*β^2*Q(PART) *Rpcs(POIo(t),t=0)^(-2)*sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART))*Q(PART)*Rpcs(POIo(t),t=0)^(-2) 1109:(mathH) E0ods(POIo,t) = E0pds(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2 (mathH)/* when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! K_1st(t=0) = E0ods(POIo,t)*3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 - E0ods(POIo,t)*λ(Vons(PART)) (endMath) ????:(mathH) K_2nd = 21/8*β^4*Q(PART)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4 - λ(Vons(PART))*β^2*Q(PART)*Rocs(POIo)*Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t=0))^2 for use in differentiations 1049:(mathH) Rocs(POIo) = Rpcs(POIo(t),t=0) when : t=0, RFp=RFo @t=0 K_2nd = 21/8*β^4*Q(PART)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4 - λ(Vons(PART))*β^2*Q(PART) *Rpcs(POIo(t),t)^(-2)*sin(Aθpc(POIo(t),t=0))^2 1109:(mathH) E0ods(POIo,t) = E0pds(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2 (mathH)/* HIGHLY restricted! [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! /% K_2nd(t=0) = E0ods(POIo,t)*21/8*β^4*Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t=0))^4 - E0ods(POIo,t)*λ(Vons(PART))*β^2*sin(Aθpc(POIo(t),t=0))^2 (endMath) /********************* >>>>>>>>> E0pdv(POIp) Gauss's Law for a single point charge, in the particle reference frame (RFp) : /% E0pdv(POIp) = Q(PART) /|Rpcv(POIp)|^2 *Rpch(POIp) = Q(PART) / Rpcs(POIp) ^2 *Rpch(POIp) (mathH) E0pdv(POIp) = Q(PART)/Rpcs(POIp)^2*Rpch(POIp) (endMath) (mathH) E0pds(POIp) = Q(PART)/Rpcs(POIp)^2 (endMath) /*Here I have not yet expressed in terms of (RFo).... /********************* >>>>>>>>> E0odv(POIo,t) /* In [Maxwell, relativity] electrodynamics, time delays are ignored!! (wrong!) Therefore, the STATIC E field is the same in RFp and RFo coordinates. It is NOT restricted to t=0 when RFp=RFo, as Rpcs(POIo(t),t) is used rather than Rocs(POIo) /% (mathH) E0odv(POIo,t) = E0pdv(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2 *Rpch(POIo(t),t) (endMath) (mathH) E0ods(POIo,t) = E0pds(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2 (endMath) /********************* >>>>>>>>> B0pdv(POIp) = B0odv(POIo) = 0 In this analysis, and implicitly in Lucas (compare pp66h0.5 Equation (4-13) with p67h0.6 Equation (4-13), it is assumed that there are NO external magnetic fields, (i.e. from other sources, independent of the particle), nor is the particle magnetic, and so B0 is a constant zero (assuming that this is the definition of B0) for all reference frames. Magnetic fields must therefore arise from the electrostatic or induced electric fields arising from the charged particle. (mathH) B0pdv(POIp) = 0 (endMath) (mathH) B0odv(POIo) = 0 (endMath) /********************* >>>>>>>>> BIpdv(POIp) = 0 ≠ BIodv(POIp(t),t) = BIodv(POIo,t) As defined here, induced magnetic fields arise from the movement of electric fields, for example as arising from a moving charged particle. In this work, only field arising from the particle and its movement are considered. However, the charged particle does NOT move with respect to the particle reference frame RFp, so its relative velocity is zero, therefore NO magnetic field arises in RFp. Therefore, there is no induced magnetic field for Points Of Interest (POIp) that are fixed in the particle reference frame (RFp). This is shown in the derivation below. Generalized Ampere's Law, in (RFp) ("X" is the vector cross-product) : /% BIpdv(POIp) = Vpnv(POIp)/c X E0pdv(Rpcv(POIp)) = 0 /c X E0pdv(Rpcv(POIp)) = 0 (mathH) BIpdv(POIp) = 0 (endMath) /********************* >>>>>>>>> BTpdv(POIp) = 0 Assuming : /% BTpdv(POIp) = B0pdv(POIp) + BIpdv(POIp) = 0 + 0 = 0 /* where (mathH) B0pdv(POIp) = 0 as given in Chapter 4 (endMath) - magnetic field external (currents, permanent mags) in (RFp) BIpdv(POIp) - magnetic field induced by charge Q(PART), which moves in RFo but does NOT move in (RFp) /% (mathH) BTpdv(POIp) = 0 (endMath) /********************* >>>>>>>>> EIpdv(POIp) = 0 The INDUCED electric field arises from CHANGES in the total magnetic field at a point. In the particle reference frame there is no change in magnetic field at POIp, hence no induced electric field. (mathH) EIpdv(POIp) = 0 (endMath) /********************* >>>>>>>>> ETpdv(POIp) = E0pdv(POIp) /% ETpdv(POIp) = E0odv(POIo,t) + EIodv(POIo,t) = Q(PART)/Rpcs(POIp)^2*Rpch(POIp) + 0 = Q(PART)/Rpcs(POIp)^2*Rpch(POIp) (mathH) ETpdv(POIp) = Q(PART)/Rpcs(POIp)^2*Rpch(POIp) (endMath) /***************************************** >>>>>> Derivatives /********************* >>>>>>>>> Figure "Calculus for RFp, using POIp=POIo(t)" Run command to see $ eog "$d_images""Howell - Chapter 4 - POIo calculus - cropped.png" & /********************* >>>>>>>>> Reminders for Chapter 4 : Vonv(PART) = constant [Particle, observer] frames of reference (RFp) & (RFo) are IDENTICAL [scale, rotation, etc] at time t=0, apart from the given that the particle's reference frame (RFp) moves with velocity Vonv(PART) relative to RFo. Reminders for (POIo) - Point Of Interest that is FIXED in the observer reference frame (RFo) : [Roc,Ooc,Poc] and their derived concepts, are NOT functions of time, i.e. (POIp). Their derivatives are zero. [Rpc,Opc,Ppc], their related concepts, and their derivatives are all functions of time, i.e. (POIp) /********************* >>>>>>>>> OBJECTIVES : Where convenient, express variables as functions of FIXED variables : (POIp) -> -Vodv(PART),t,Rpcv(POIp),Aθpc(POIp),Aφpc(POIp) (POIo) -> Vodv(PART),t,Rocv(POIo),Aθoc(POIo),Aφoc(POIo) 28Sep2019 - nah, not so importnat! I just want answers /********************* >>>>>>>>> ∂[∂(t): Rpcv(POIp)] = ∂[∂(t): Aθpc(POIp)] = 0 As per "Rpcv(POIp), Aθpc(POIp), Ppc(POIp)" above, Rpc and Opc are constants for a given POIp, so their derivatives are zero : /% (mathH) ∂[∂(t): Rpcv(POIp)] = 0 (endMath) (mathH) ∂[∂(t): Aθpc(POIp)] = 0 (endMath) /********************* >>>>>>>>> ∂[∂(t): Rpcv(POIo(t) ] From Galilaean invariance, Eqn (1) in "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)" fixed in RFo space : /% Rpcv(POIo(t),t) = Rocv(POIo) - Vonv(PART)*t /*Meaning that for any POIo : /% ∂[∂(t): Rpcv(POIo(t),t) = ∂[∂(t): Rocv(POIo) - Vonv(PART)*t] = ∂[∂(t): Rocv(POIo)] - ∂[∂(t): Vonv(PART)*t] = 0 - Vonv(PART) (mathH) ∂[∂(t): Rpcv(POIo(t),t)] = -Vonv(PART) (endMath) /*This is expected. /*+-----+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to length/time /********************* >>>>>>>>> Figure "∂[∂(t): Rpcs(POIo(t),t) ]" Run command to see $ eog "$d_images""Howell - Chapter 4 - dp[dt _ Rpcs(POIo,t)] - cropped.png" & /*$ cat >>"$p_augmented" "$d_augment""d-dt Rpcs.txt" /*_endCmd /********************* >>>>>>>>> ∂[∂(t): Rpcs(POIo(t),t)] /% ∂[∂(t): Rpcs(POIo(t),t) ] = ∂[∂(t): |Rpcv(POIo(t),t)|] Rpcs(POIo(t),t) = |Rpcv(POIo(t),t)| so ∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|] /********************* >>>>>>>>>>>> Kahan formulation for derivatives of magnitudes : d||z|| = u_T dotProd dz / ||z|| where u_T is the linear functional dual to z wrt ||...|| HERE I INTERPRET u_T = Rpcv(POIo(t),t) /% /*$ cat >>"$p_augmented" "$d_augment""Kahan formulation for derivatives of magnitudes.txt" /*_endCmd /********************* >>>>>>>>> ∂[∂(t): Rpcs(POIo(t),t)^(-α)] /% ∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|] = (-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) /* example /% ∂[∂(t): Rpcs(POIo(t),t)^(-3)] = (-3)*Rpcs(POIo(t),t)^(-4)*∂[∂(t): Rpcs(POIo(t),t)] = (-3)*Rpcs(POIo(t),t)^(-4)*(-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) = 4*Vons(PART)*Rpcs(POIo(t),t)^(-4)*cos(Aθpc(POIo(t),t)) /* pattern /% ∂[∂(t): Rpcs(POIo(t),t)^(-α)] = (-α)*Rpcs(POIo(t),t)^(-α - 1)*∂[∂(t): Rpcs(POIo(t),t)] = (-α)*Rpcs(POIo(t),t)^(-α - 1)*(-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) (mathH) ∂[∂(t): Rpcs(POIo(t),t)^(-α)] = α*Vons(PART)*Rpcs(POIo(t),t)^(-α - 1)*cos(Aθpc(POIo(t),t)) (endMath) /********************* >>>>>>>>> Figure "∂[∂(t): Aθpc(POIo(t),t)]" Run command to see $ eog "$d_images""Howell - Chapter 4 - dp[dt _ Rpcs(POIo,t)] - cropped.png" & /********************* >>>>>>>>> ∂[∂(t): Aθpc(POIo(t),t)] /*$ cat >>"$p_augmented" "$d_augment""d-dt Aθpc.txt" /*_endCmd /********************* >>>>>>>>> Figure "∂[∂(t): Rpch(POIo(t),t)]" http://www.BillHowell.ca/ /media/bill/SWAPPER/Lucas - Universal Force/Images/Howell - Chapter 4 - dp[dt _ Rpch(POIo,t)] - cropped.png /********************* >>>>>>>>> ∂[∂(t): Rpch(POIo(t),t) ] see Figure "∂[∂(t): Rpch(POIo(t),t)]" Rpch(POIo(t),t) = vector[length = 1, theta = Aθpc(POIo(t),t), phi = Aφpc(POIo) = constant ] So : (1) d[Rpch(POIo(t),t)] = d[Aθpc(POIo(t),t)]*1*RDEpdh(POIo(t),t) where RDEpdh(POIo(t),t) is anchored at end of Rpch(POIo(t),t) and is at angle Aθpc(POIo(t),t) + PI/2, ie perpendicular to Rpch(POIo(t),t) angle Aφpc(POIo(t),t) doesn't change /*+-----+ (RFp) basis From "∂[∂(t): Aθpc(POIo(t),t)]" : (3) ∂[∂(t): Aθpc(POIo(t),t)] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) Subbing (3)* into (1) : (1) ∂[∂(t): Rpch(POIo(t),t)] = ∂[∂(t): Aθpc(POIo(t),t)]*1*RDEpdh(POIo(t),t) = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*RDEpdh(POIo(t),t) Summarizing : (mathH) ∂[∂(t): Rpch(POIo(t),t)] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*RDEpdh(POIo(t),t) (endMath) where RDEpdh(POIo(t),t) is anchored at end of Rpch(POIo(t),t) and is at angle Aθpc(POIo(t),t) + PI/2, ie perpendicular to Rpch(POIo(t),t) 23Jan2017 changed to PLUS PI/2? (was that way in half of the expressions - maybe should be for RFp? angle Aφpc(POIo(t),t) doesn't change /*+-----+ (RFo) basis From "Rpcs(POIo(t),t)" : (3) Rpcs(POIo(t),t) = |Rpcv(POIo(t),t)| = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) From "sin(Aθpc(POIo(t),t))" : (5) sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (3)* & (5)* into (2) : (2) ∂[∂(t): Rpch(POIo(t),t)] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*RDEpdh(POIo(t),t) = Vons(PART)*RDEpdh(POIo(t),t) *sin(Aθpc(POIo(t),t)) /Rpcs(POIo(t),t) = Vons(PART)*RDEpdh(POIo(t),t) * Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Finally : (mathH)/* where RDEpdh(POIo(t),t) is anchored at end of Rpch(POIo(t),t), is at angle Aθpc(POIo(t),t) + PI/2, ie perpendicular to Rpch(POIo(t),t), angle Aφpc(POIo(t),t) doesn't change /% ∂[∂(t): Rpch(POIo(t),t)] = Vons(PART)*RDEpdh(POIo(t),t)*Rocs(POIo)*sin(Aθoc(POIo)) / {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2} (endMath) /*+-----+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to 1/time (angles & unit vectors are dimensionless) /********************* >>>>>>>>> ∂[∂(t): sin(Aθpc(POIo(t),t))] /*MUCH LATER : use (4-17) 'Sperical coordinate transforms ' WARNING : Should do vector differentiation THEN take the magnitude!?? NOTE : Equation numbering is "weird", as I first did the "RFo" approach, whereas usually I start with the RFp approach. In consequence RFp equations start with the number "10". /*$ cat >>"$p_augmented" "$d_augment""d-dt sin.txt" /*_endCmd /********************* >>>>>>>>> ∂[∂(t): cos(Aθpc(POIo(t),t))] Reminders for Chapter 4 : Vonv(PART) = constant [Particle, observer] frames of reference (RFp) & (RFo) are IDENTICAL [scale, rotation, etc] at time t=0, apart from the given that the particle's reference frame (RFp) moves with velocity Vonv(PART) relative to RFo. Reminders for (POIo) - Point Of Interest that is FIXED in the observer reference frame (RFo) : [Roc,Ooc,Poc] and their derived concepts, are NOT functions of time, i.e. (POIp). Their derivatives with respect to time are zero. [Rpc,Opc,Ppc], their related concepts, and their derivatives are all functions of time, i.e. (POIp) /*$ cat >>"$p_augmented" "$d_augment""d-dt cos.txt" /*_endCmd /********************* >>>>>>>>> ∂[∂(t): Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))] = 0 WARNING : Should do vector differentiation THEN take the magnitude!?? Looking at Figure "∂[∂(t): Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]", it is clear by inspection that : /% /*$ cat >>"$p_augmented" "$d_augment""Rpcs*sin.txt" /*_endCmd /********************* >>>>>>>>> ∂[∂(t): Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))] WARNING : Should do vector differentiation THEN take the magnitude!?? Looking at Figure "Basic measures for a POIo", it is clear by inspection that : /% /*$ cat >>"$p_augmented" "$d_augment""d-dt Rpcs*_cos.txt" /*_endCmd /********************* >>>>>>>>> ∂[∂(t): Rpcs(POIo(t),t)^(-b)*sin(Aθpc(POIo(t),t))^a] 28Sep2019 - just started fixing for ∂[∂(t): Aθpc(POIo(t),t)] this is a HUGE CHANGE, much simpler than all previous work!!! to generate (mathH) lines : see "Binomial Series for Chapter 4.ods" /*$ cat >>"$p_augmented" "$d_augment""d-dt Rpcs^-b*sin^a.txt" /*_endCmd /********************* >>>>>>>>> ∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] 24Sep2019 initial versions 16Oct2019 file_inserts for two cases /% +-----+ /*$ cat >>"$p_augmented" "$d_Lucas""math Howell/cos - 1 noo, iterative, non-feedback/d-dt Rpcs^-5*t*_cos - 1.txt" /*_endCmd +-----+ /*$ cat >>"$p_augmented" "$d_Lucas""math Howell/cos - 1 yes, iterative, non-feedback/d-dt Rpcs^-5*t*_cos - 1.txt" /*_endCmd /********************* >>>>>>>>> ∂[∂(t): K0,K2] for use in differentiations /% 1005:(mathH)# for use in differentiations K0 = 3/2*β^2*Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2 ∂[∂(t): K0] = ∂[∂(t): 3/2*β^2*Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2] = 3/2*β^2*Rocs(POIo)^3*Q(PART) *∂[∂(t): Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2] 2121:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-5)] = 7*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) = 3/2*β^2*Rocs(POIo)^3*Q(PART) *7*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) (mathH)/* for use in differentiations /% ∂[∂(t): K0] = 21/2*β^2*Rocs(POIo)^3*Q(PART)*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) (endMath) 1006:(mathH) K2 = (-1)*λ(Vons(PART))*Q(PART)*Rpcs(POIo(t),t)^(-2) for use in differentiations ∂[∂(t): K2] = ∂[∂(t): (-1)*λ(Vons(PART))*Q(PART)*Rpcs(POIo(t),t)^(-2)] = (-1)*λ(Vons(PART))*Q(PART) *∂[∂(t): Rpcs(POIo(t),t)^(-2)] 1417:(mathH) ∂[∂(t): Rpcs(POIo(t),t)^(-α)] = α*Vons(PART)*Rpcs(POIo(t),t)^(-α - 1)*cos(Aθpc(POIo(t),t)) = (-1)*λ(Vons(PART))*Q(PART) *2 *Vons(PART)*Rpcs(POIo(t),t)^(-2 - 1)*cos(Aθpc(POIo(t),t)) (mathH)/* for use in differentiations /% ∂[∂(t): K2] = (-2)*λ(Vons(PART))*Q(PART)*Vons(PART)*Rpcs(POIo(t),t)^(-3)*cos(Aθpc(POIo(t),t)) (endMath) /********************* >>>>>>>>> ∂[∂(t): K1] for use in differentiations From "K0, K1, K2, K3 for integro-differential equations" 03Oct2019 I need to check this before using it!! /% K1 = -3*β^2*Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(-5)*Vons(PART)*t*(cos(Aθpc(POIo(t),t=0)) - 1) ∂[∂(t): K1] = ∂[∂(t): 3*β^2 *Rocs(POIo)^2*Rpcs(POIo(t),t)^(-5)*Q(PART)*(-1)*Vons(PART)*t*(cos(Aθpc(POIo(t),t)) - 1) ] = -3*β^2*Vons(PART)*Rocs(POIo)^2*Q(PART)*∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1) ] /* insert derivative expressions from "Howell - Background math for Lucas Universal Force, Chapter 4.txt" section "∂[∂(t): Rpcs(POIo(t),t)^(-b)*sin(Aθpc(POIo(t),t))^a]" section "∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)]" /% (mathH)/* ???? where is the section on this? /% ∂[∂(t): K1] = -3*β^2*Vons(PART)*Rocs(POIo)^2*Q(PART) * 0 (endMath) /* Note the zeroing of the derivative term under several HIGHLY restrictive conditions!!! - RFp = RFo, particle is at origin (both reference frames) - for a POIo along direction of flight of particle, so cos(Aθpc(POIo(t),t)) = 1 - t=0 therefore : /% (mathH)/* when [t=0, RFp=RFo @t=0], maybe use only AFTER differentiations??? /% ∂[∂(t): K1(t=0)] = 0 (endMath) /********************* >>>>>>>>> ∂[∂(t): K0(t=0),K2(t=0)] in terms of E0ods(POIo,t) for relativistic factor, when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! /% ????:(mathH) ∂[∂(t): K0] = 21/2*β^2*Rocs(POIo)^3*Q(PART)*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) for use in differentiations ∂[∂(t): K0(t=0)] = 21/2*β^2*Rocs(POIo)^3*Q(PART)*Vons(PART)*sin(Aθpc(POIo(t),t=0))^2*cos(Aθpc(POIo(t),t=0))*Rpcs(POIo(t),t=0)^(-6) 1049:(mathH) Rocs(POIo) = Rpcs(POIo(t),t=0) when : t=0, RFp=RFo @t=0 = 21/2*β^2*Q(PART) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) 1118:(mathH) E0ods(POIo,t) = E0pds(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2 = E0ods(POIo,t) *21/2*β^2*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-1) (mathH)/* when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! /% ∂[∂(t): K0(t=0)] = E0ods(POIo,t)*21/2*β^2*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-1) (endMath) ????:(mathH) ∂[∂(t): K2] = (-2)*λ(Vons(PART))*Q(PART)*Vons(PART)*Rpcs(POIo(t),t)^(-3)*cos(Aθpc(POIo(t),t)) for use in differentiations ∂[∂(t): K2(t=0)] = (-2)*λ(Vons(PART))*Q(PART)*Vons(PART)*Rpcs(POIo(t),t=0)^(-3)*cos(Aθpc(POIo(t),t=0)) 1118:(mathH) E0ods(POIo,t) = E0pds(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2 = (-2)*λ(Vons(PART)) *Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIo(t),t=0)) (mathH)/* when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! /% ∂[∂(t): K2(t=0)] = (-2)*λ(Vons(PART))*Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIo(t),t=0)) (endMath) /********************* >>>>>>>>> ∂[∂(t): K_1st,K_2nd,K_3rd] for use in differentiations 1042:(mathH) K_1st = 3/2*β^2*Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t=0)^(-5)*sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART))*Q(PART)*Rpcs(POIo(t),t=0)^(-2) for use in differentiations ∂[∂(t): K_1st] = ∂[∂(t): 3/2*β^2*Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t=0)^(-5)*sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART))*Q(PART)*Rpcs(POIo(t),t=0)^(-2)] = + ∂[∂(t): 3/2*β^2*Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t=0)^(-5)*sin(Aθpc(POIo(t),t=0))^2] - ∂[∂(t): λ(Vons(PART))*Q(PART)*Rpcs(POIo(t),t=0)^(-2)] /* percolate constants /% = + 3/2*β^2*Q(PART)*Rocs(POIo)^3 *∂[∂(t): Rpcs(POIo(t),t=0)^(-5)*sin(Aθpc(POIo(t),t=0))^2] - λ(Vons(PART))*Q(PART) *∂[∂(t): Rpcs(POIo(t),t=0)^(-2)] 2110:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-5)] = 7*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) 2100:(mathH) ∂[∂(t): Rpcs(POIo(t),t)^(-2)] = 2*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) = + 3/2*β^2*Q(PART)*Rocs(POIo)^3 *7*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) - λ(Vons(PART))*Q(PART) *2*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) (mathH)/* for use in differentiations /% ∂[∂(t): K_1st] = 21/2*β^2*Q(PART)*Rocs(POIo)^3*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) - 2*λ(Vons(PART))*Q(PART)*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) (endMath) /* Check /% 1038:(mathH) K_1st = K0 + K2 for use in differentiations /* therefore /% ∂[∂(t): K_1st] = ∂[∂(t): K0 + K2] = ∂[∂(t): K0] + ∂[∂(t): K2] 2389:(mathH) ∂[∂(t): K0] = 21/2*β^2*Rocs(POIo)^3*Q(PART)*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) for use in differentiations 2398:(mathH) ∂[∂(t): K2] = (-2)*λ(Vons(PART))*Q(PART)*Vons(PART)*Rpcs(POIo(t),t)^(-3)*cos(Aθpc(POIo(t),t)) for use in differentiations = 21/2*β^2*Rocs(POIo)^3*Q(PART)*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) + (-2)*λ(Vons(PART))*Q(PART)*Vons(PART)*Rpcs(POIo(t),t)^(-3)*cos(Aθpc(POIo(t),t)) = 21/2*β^2*Rocs(POIo)^3*Q(PART)*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) - 2*λ(Vons(PART))*Q(PART)*Vons(PART)*Rpcs(POIo(t),t)^(-3)*cos(Aθpc(POIo(t),t)) = 21/2*β^2*Q(PART)*Rocs(POIo)^3*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) - 2*λ(Vons(PART))*Q(PART)*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) /* OK - this works /% 1063:(mathH) K_2nd = 21/8*β^4*Q(PART)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4 - λ(Vons(PART))*β^2*Q(PART)*Rocs(POIo)*Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t=0))^2 for use in differentiations ∂[∂(t): K_2nd] = ∂[∂(t): 21/8*β^4*Q(PART)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4 - λ(Vons(PART))*β^2*Q(PART)*Rocs(POIo)*Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t=0))^2] = + ∂[∂(t): 21/8*β^4*Q(PART)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4] - ∂[∂(t): λ(Vons(PART))*β^2*Q(PART)*Rocs(POIo)*Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t=0))^2] /* percolate constants /% = + 21/8*β^4*Q(PART) *∂[∂(t): Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4] - λ(Vons(PART))*β^2*Q(PART)*Rocs(POIo) *∂[∂(t): Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t=0))^2] 2114:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-6)] = 10*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) 2106:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-3)] = 5*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) = + 21/8*β^4*Q(PART) *10*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) - λ(Vons(PART)) *β^2*Q(PART)*Rocs(POIo) *5*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) = + 210/8*β^4*Q(PART) *Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) - λ(Vons(PART))*5*β^2*Q(PART)*Rocs(POIo) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) (mathH)/* for use in differentiations /% ∂[∂(t): K_2nd] = + 210/8*β^4*Q(PART) *Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) - λ(Vons(PART))*5*β^2*Q(PART)*Rocs(POIo) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) (endMath) /********************* >>>>>>>>> ∂[∂(t): K_1st(t=0),K_2nd(t=0),K_3rd(t=0)] in terms of E0ods(POIo,t) for relativistic factor, when [t=0, RFp=RFo @t=0], use only AFTER differentiations!!! /% 2468:(mathH) ∂[∂(t): K_1st] = 21/2*β^2*Q(PART)*Rocs(POIo)^3*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) - 2*λ(Vons(PART))*Q(PART)*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) for use in differentiations ∂[∂(t): K_1st(t=0)] = 21/2*β^2*Q(PART)*Rocs(POIo)^3*Vons(PART)*sin(Aθpc(POIo(t),t=0))^2*cos(Aθpc(POIo(t),t=0))*Rpcs(POIo(t),t=0)^(-6) - λ(Vons(PART))*2*Q(PART)*Vons(PART)*cos(Aθpc(POIo(t),t=0))*Rpcs(POIo(t),t=0)^(-3) 1049:(mathH) Rocs(POIo) = Rpcs(POIo(t),t=0) when : t=0, RFp=RFo @t=0 = 21/2*β^2 *Q(PART)*Vons(PART)*sin(Aθpc(POIo(t),t=0))^2*cos(Aθpc(POIo(t),t=0))*Rpcs(POIo(t),t=0)^(-3) - λ(Vons(PART))*2*Q(PART)*Vons(PART) *cos(Aθpc(POIo(t),t=0))*Rpcs(POIo(t),t=0)^(-3) 1109:(mathH) E0ods(POIo,t) = E0pds(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2 = E0ods(POIo,t) *21/2*β^2 *Vons(PART)*sin(Aθpc(POIo(t),t=0))^2*cos(Aθpc(POIo(t),t=0))*Rpcs(POIo(t),t=0)^(-1) - E0ods(POIo,t)*λ(Vons(PART)) *2 *Vons(PART) *cos(Aθpc(POIo(t),t=0))*Rpcs(POIo(t),t=0)^(-1) (mathH)/* when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! /% ∂[∂(t): K_1st(t=0)] = E0ods(POIo,t) *21/2*β^2*Vons(PART)*sin(Aθpc(POIo(t),t=0))^2*cos(Aθpc(POIo(t),t=0))*Rpcs(POIo(t),t=0)^(-1) - E0ods(POIo,t)*λ(Vons(PART)) *2 *Vons(PART) *cos(Aθpc(POIo(t),t=0))*Rpcs(POIo(t),t=0)^(-1) (endMath) 2497:(mathH) ∂[∂(t): K_2nd] = 210/8*β^4*Q(PART)*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) - λ(Vons(PART))*5*β^2*Q(PART)*Rocs(POIo)*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) for use in differentiations ∂[∂(t): K_2ndt=0)] >> 04Oct2019 wait for later - I might not need this? 1049:(mathH) Rocs(POIo) = Rpcs(POIo(t),t=0) when : t=0, RFp=RFo @t=0 1109:(mathH) E0ods(POIo,t) = E0pds(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2 /********************* >>>>>>>>> ∂[∂(t): E0pds(POIp)] using /% 1065:(mathH) E0pds(POIp) = Q(PART)/Rpcs(POIp)^2 /* Because there is no change in the static electric field in RFp coordinates : /% (mathH) ∂[∂(t): E0pds(POIp)] = 0 (endMath) /********************* >>>>>>>>> ∂[∂(t): E0ods(POIo,t)] In [Maxwell, relativity] electrodynamics, time delays are ignored!! (wrong!) Therefore, the STATIC E field is the same in RFp and RFo coordinates. /* using /% 1077:(mathH) E0ods(POIo,t) = E0pds(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2 ∂[∂(t): E0ods(POIo,t)] = ∂[∂(t): Q(PART)/Rpcs(POIo(t),t)^2] = Q(PART) *∂[∂(t): Rpcs(POIo(t),t)^(-2)] 1377:(mathH) ∂[∂(t): Rpcs(POIo(t),t)^(-α)] = α*Vons(PART)*Rpcs(POIo(t),t)^(-α - 1)*cos(Aθpc(POIo(t),t)) = Q(PART)*α*Vons(PART)*Rpcs(POIo(t),t)^(-α - 1)*cos(Aθpc(POIo(t),t)) = Q(PART)*2*Vons(PART)*Rpcs(POIo(t),t)^(-2 - 1)*cos(Aθpc(POIo(t),t)) (mathH) ∂[∂(t): E0ods(POIo,t)] = 2*Q(PART)*Vons(PART)*Rpcs(POIo(t),t)^(-3)*cos(Aθpc(POIo(t),t)) (endMath) /* putting into form with E0ods(POIo,t), again using /% 1077:(mathH) E0ods(POIo,t) = E0pds(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2 (mathH) ∂[∂(t): E0ods(POIo,t)] = E0ods(POIo,t) *2*Vons(PART)*Rpcs(POIo(t),t)^(-1)*cos(Aθpc(POIo(t),t)) (endMath) /********************* >>>>>>>>> Summary of ith stage EIods(POIo,t,ith stage)) derivations /*$ cat >>"$p_augmented" "$d_augment""d-dt EIods - summary of ith stages.txt" /*_endCmd /********************* >>>>>>>>> Summary of ith stage ETods(POIo,t,ith stage)) The change from EIods(POIo,t,ith stage)) to ETods(POIo,t,ith stage)) is simple - just add a "one" to the (β*sin)^n series for E0pds(POIp) (1st expression RHS). Example for the 2nd stage is given below. /% 3016:(mathL)/* HIGHLY restrictive conditions : t=0, RFo(t=0) = Rfp(t=0), Rocs(POIo) = Rpcs(POIo(t),t=0) This means that the [observer, particle] reference frames are exactly the same at t=0 (other than motion). drop as roundoff error : f_sphereCapSurf expression see "textIn - relativistic factor, intermediate symbols, restrictive conditions.txt" /% EIods(POIo,t=0,2nd stage) = E0ods(POIo,t) *{ 3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 + 21/8*β^4*sin(Aθpc(POIo(t),t=0))^4} - E0ods(POIo,t)*λ(Vons(PART))*{1 + β^2*sin(Aθpc(POIo(t),t=0))^2} 3026:(mathL)/* HIGHLY restrictive conditions : t=0, RFo(t=0) = Rfp(t=0), Rocs(POIo) = Rpcs(POIo(t),t=0) This means that the [observer, particle] reference frames are exactly the same at t=0 (other than motion). drop as roundoff error : f_sphereCapSurf expression see "textIn - relativistic factor, intermediate symbols, restrictive conditions.txt" /% ETods(POIo,t=0,2nd stage) = E0ods(POIo,t) *{1 + 3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 + 21/8*β^4*sin(Aθpc(POIo(t),t=0))^4} - E0ods(POIo,t)*λ(Vons(PART))*{1 + β^2*sin(Aθpc(POIo(t),t=0))^2} /********************* >>>>>>>>> ∂[∂(t): K_1st + f_sphereCapSurf(EIods(POIo,t=0,(i-1) stage))) ] >>>>>>>>>>>> General process 1. derive ∂[∂(t): f_sphereCapSurf(EIods(POIo,t,(i-1) stage))) ] 2. express full K_1st + f_sphereCapSurf(EIods(POIo,t=0,(i-1) stage))) using result from 1 above and standard ∂[∂(t): K_1st] 2669:(mathH)/* for use in differentiations /% ∂[∂(t): K_1st] = 21/2*β^2*Q(PART)*Rocs(POIo)^3*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) - 2*λ(Vons(PART))*Q(PART)*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) 3. adjust to get integrable form by substiting for t=0 to get ∂[∂(t): K_1st + f_sphereCapSurf(EIods(POIo,t=0,(i-1) stage))) ] ********************* >>>>>>>>> ∂[∂(t): E0ods(POIo,t)*sin(Aθpc(POIo(t),t))^a] re-checked 03Oct2019 /*$ cat >>"$p_augmented" "$d_augment""d-dt E0ods*sin^a.txt" /*_endCmd /********************* >>>>>>>>> ∂[∂(t): E0ods(POIo,t=0)*Rpcs(POIo(t),t)^(-β)*sin(Aθoc(POIo))^a] 03Oct2019 - this is very old and must be [checked, modified] if it is to be used!! get rid of all "Aθtc" -> Aθ[o,p]c 08Jun2016 - Can I prove that the term with Vons << the second term, so that it may be dropped? (I doubt it very much!! except for very special conditions!) - 1/c^n terms does it, but makes Lucas's work irrelevant. - I must have screwed up again somewhere... /% /*$ cat >>"$p_augmented" "$d_augment""d-dt E0ods*Rpcs^-β*sin^a.txt" /*_endCmd /********************* >>>>>>>>> ∂[∂(t): E0pdv(POIp) ] = ∂[∂(t): E0pdv(POIo(t),t) ] = 0 ≠ ∂[∂(t): E0odv(POIo)] From "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt", section ""Points of Interest" (POI)s", normally, E0pdv(POIp) and E0pdv(POIo(t),t) are NOT interchangeable, as the latter refers to a trajectory of a POIo over time as seen in the observer reference frame RFo. However, it is shown here in the context that within the particle frame of reference, E0, and therefore [BI, BT, EI, ET] do not change. As the static electric field (electrostatic field) in RFp is constant : /%(1) ∂[∂(t): E0pdv(POIo(t),t)] = 0 /********************* >>>>>>>>> ∂[∂(t): BIpdv(POIp) ] = ∂[∂(t): BIpdv(POIo(t),t) ] = ∂[∂(t): BTpdv(POIo(t),t) ] = 0 ≠ ∂[∂(t): BTodv(POIo,t)] From "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt", section ""Points of Interest" (POI)s", normally, BIpdv(POIp) and E0pdv(POIo(t),t) are NOT interchangeable, as the latter refers to a trajectory of a POIo over time as seen in the observer reference frame RFo. However, it is shown here in the context that within the particle frame of reference RFp, E0, and therefore [BI, BT, EI, ET] do not change, so BIpdf(POIp) and BTpdf(POIp) are NOT functions of time, and are equal. For the Chapter 4 situation, there is no "external" magnetic field (independent of the particle), and so B0 is a constant zero. As per the previous sub-section, the induced magnetic field is always zero, which means that its derivative is also zero in RFp. /%(1) ∂[∂(t): BIpdv(POIo(t),t)] = ∂[∂(t): BTpdv(POIo(t),t)] = 0 /*This is a strange comment - not one that I normally think of... endsection /********************************************** >>>>>> Integrals /********************* >>>>>>>>> ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^z] /% /*$ cat >>"$p_augmented" "$d_augment""∫_d_Aθpc, cos*sin^z.txt" /*_endCmd /********************* >>>>>>>>> ∫[∂(Aθpc),0.to.Aθpcf: Rpcs(POIo(t),t)^(-β)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a)] 30Aug2019 This section is probably irrelevant in light of the conclusions of : /* 30Aug2019 see "Howell - Background math for Lucas Universal Force, Chapter 4.txt" section '"Rpcs(POIo(t),t)" is a constant for integrals like "∫{∂(Aθpc),0.to.Aθoc(POIp(t),t=0):" ?' Rpcs(POIo(t),t=0) is a constant with respect ot that type of integral. /*$ cat >>"$p_augmented" "$d_augment""∫d_Aθpc, Rpcs^-β*cos*sin^a.txt" /*_endCmd /********************************************** >>> III. Derivations for a POIo = POIp(t) fixed in the observer reference frame (RFo) /***************************************** >>>>>> Basic measures /********************* >>>>>>>>> Figure "Basic measures for the observer reference frame RFo, using POIo=POIp(tx)" http://www.BillHowell.ca/ /media/bill/SWAPPER/Lucas - Universal Force/Images/Howell - Chapter 4 - POIo basic - cropped.png Reminders for Chapter 4 : Vonv(PART) = constant [Particle, observer] frames of reference (RFp) & (RFo) are IDENTICAL [scale, rotation, etc] at time t=0, apart from the given that the particle's reference frame (RFp) moves with velocity Vonv(PART) relative to RFo. Reminders for (POIo) - Point Of Interest that is FIXED in the observer reference frame (RFo) : [Roc,Ooc,Poc] and their derived concepts, are NOT functions of time, i.e. (POIp). Their derivatives with respect to time are zero. [Rpc,Opc,Ppc], their derived concepts, and their derivatives are all functions of time, i.e. (POIp) OBJECTIVES : Where convenient, express variables as functions of FIXED variables : (POIp) -> -Vodv(PART),t,Rpcv(POIp),Aθpc(POIp),Aφpc(POIp) (POIo) -> Vodv(PART),t,Rocv(POIo),Aθoc(POIo),Aφoc(POIo) /********************* >>>>>>>>> Rocv(POIo), Aθoc(POIo), Aφoc(POIo) are constants in RFo As per the introduction to this section, : This section deals with (POIo) - Points Of Interest(POI) which are fixed with respect to the observer reference frame (RFo). In other words, the (POIo) is stationary with respect to the observer and moves with constant relative velocity -Vonv(PART) with respect to the particle. /% (mathH)/* at time t when POIo and POIp(t) are the same point /% Rocv(POIo) = constant ≠ Rpcv(POIo(t),t) (mathH)/* at time t when POIo and POIp(t) are the same point /% Aθoc(POIo) = constant ≠ Aθpc(POIo(t),t) (endMath) (mathH)/* at time t when POIo and POIp(t) are the same point /% Aφoc(POIo) = constant ≠ Aφpc(POIo(t),t) (endMath) /********************* >>>>>>>>> Rocv(POIp(t),t) /*+-----+ Galilean transformation : /% (mathH) Rocv(POIp(t),t) = Rpcv(POIp) + Vonv(PART)*t (endMath) /*Limit checks : Dimensional consistency - OK, as all terms reduce to (length). /********************* >>>>>>>>> Rocs(POIp(t),t) Distance Rocs(POIp(t),t) from the (RFo) origin : /% Rocs(POIp(t),t) = |Rpcv(POIp) + Vonv(PART)*t| = { [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 }^(1/2) = { [Rpcs(POIp)*cos(Aθpc(POIo(t),t))]^2 + 2*Rpcs(POIp)*cos(Aθpc(POIp)*Vons(PART)*t + [Vons(PART)*t]^2 + [Rpcs(POIp)*sin(Aθpc(POIp))]^2 }^(1/2) = { Rpcs(POIp)^2 *[ cos(Aθpc(POIo(t),t))^2 + sin(Aθpc(POIp))^2 ] + 2*Rpcs(POIp)*cos(Aθpc(POIp)*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /*as sin^2 + cos^2 = 1 : /%(2) (mathH) Rocs(POIp(t),t) = { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) (endMath) /*Limit checks : Dimensional consistency - OK, as all terms reduce to (length). /********************* >>>>>>>>> RO0ocs(POIp(t),t) See Figure "Basic measures for a POIp". Distance of RO0ocs(POIp(t),t) from the (RFo) origin in O0ch direction (i.e. along L(PART)) : /% (mathH) Rθ0ocs(POIp(t),t) = Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t (endMath) /*Limit checks : Dimensional consistency - OK, as all terms reduce to (length). /********************* >>>>>>>>> ROPI2ocs(POIp(t),t) = constant See Figure "Basic measures for a POIp". Notice that the distance of ROPI2pcs(POIp) = ROPI2ocs(POIp(t),t) from L(PART) in Ppch=Poch direction (i.e. perpendicular to L(PART)) is a CONSTANT : /% ROPI2ocs(POIp(t),t) = Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t)) = Rpcs(POIp)*sin(Aθpc(POIp)) (mathH) ROPI2ocs(POIp(t),t) = Rpcs(POIp)*sin(Aθpc(POIp)) (endMath) /*Limit checks : Dimensional consistency - OK, as all terms reduce to (length). /********************* >>>>>>>>> sin(Aθoc(POIp(t),t)) From (3)&(2) : /% sin(Aθoc(POIp(t),t)) = Rpcs(POIp)*sin(Aθpc(POIp)) / Rocs(POIp(t),t) = Rpcs(POIp)*sin(Aθpc(POIp)) /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) (mathH) sin(Aθoc(POIp(t),t)) = Rpcs(POIp)*sin(Aθpc(POIp)) /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) (endMath) /*Limit checks : Dimensional consistency - OK, as all terms reduce to (dimensionless). /********************* >>>>>>>>> cos(Aθoc(POIp(t),t)) /% cos(Aθoc(POIp(t),t)) = Rθ0ocs(POIp(t),t) / Rocs(POIp(t),t) Subbing (5)&(2) : cos(Aθoc(POIp(t),t)) = [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] /{Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) (mathH) cos(Aθoc(POIp(t),t)) = [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / {Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) (endMath) /*Limit checks : Dimensional consistency - OK, as all terms reduce to (dimensionless). Limit check sin^2 + cos^2 = 1 /% sin(Aθoc(POIp(t),t))^2 + cos(Aθoc(POIp(t),t)) = [ Rpcs(POIp)*sin(Aθpc(POIp)) /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ]^2 +[ [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ]^2 = { [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ]^2 +[ Rpcs(POIp)*sin(Aθpc(POIp)) ]^2 } /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } = { [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) ]^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t +[ Vons(PART)*t ]^2 +[ Rpcs(POIp)*sin(Aθpc(POIp)) ]^2 } /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } = { Rpcs(POIp)^2*[cos(Aθpc(POIo(t),t))^2 + sin(Aθpc(POIp))^2 ] + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t +[ Vons(PART)*t ]^2 } /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } = { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t +[ Vons(PART)*t ]^2 } /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } = { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } = 1 /*OK - this agrees with earlier result (1) /********************* >>>>>>>>> R_O0_ocs(POIo) (mathH) R_O0_ocs(POIo) = Rocs(POIo)*cos(Aθocs(POI)) (endMath) ... need to do R_O0_ocs(POIp(t),t) /********************* >>>>>>>>> E0odv(POIo,t) = E0pdv(POIo(t),t) ≠ E0pdv(POIp) = constant, except when t = tx The electrostatic field at a fixed point POIo in observer space RFo is a function of time. The reference frame does NOT affect the measured electroSTATIC field at a point at the instant "t = tx" when POIo and POIp coincide, i.e. it is the same as seen from the particle frame of reference RFp as it is for the observer frame RFo (or any) frame at that instant in time, at a common point in space. This CONTRASTS to the INDUCED electric field EI, as discussed in a section3 below. Furthermore, for this work time delays for fields are ignored, such that the field MOVES WITH THE PARTICLE INSTANTANEOUSLY. /*+-----+ (RFp) basis Gauss's Law for a single point charge, in the particle reference frame (RFp) : /% (mathH) E0odv(POIo,t) = E0pdv(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) (endMath) /*+-----+ (RFo) basis /*From "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)" : /% (2) Rpcs(POIo(t),t) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /*Delete as not used here : /% (6) cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) /*Subbing (2)* & (6)* into (1)* : /%(3) E0odv(POIo,t) = E0pdv(POIp) = Q(PART)/Rpcs(POIp)^2*Rpch(POIp) = Q(PART) *Rpch(POIp) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^2 = Q(PART) *Rpch(POIp) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } Finally : (mathH) E0odv(POIo,t) = E0pdv(POIo(t),t) = Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } (endMath) /*As another different description of the unit displacement vector : Rpch(POIp(t),t) => Rodh(POIo) = unit-length displacement vector [start : RFo origin; end : POIo; length : 1] or [apex : ; theta : arccos(Aθpc(POIo(t),t)), phi : Aφpc] Note that : /% Rocs(POIo)*cos(Aθoc(POIo)) = Rθ0ocs(POIo) /*+--+ LIMIT CHECKS : Dimensionality check : OK as all units reduce to (charge/length^2) Later .... Dimensional consistency in SI units with : mu - magnetic permittivity epsilon - electric permeability Other checks? /********************* >>>>>>>>> E0ods(POIo,t) = E0pds(POIo(t),t) ≠ (for t ≠ tx) E0pds(POIp) = constant /%At t = tx, then POIo & POIp=POIo(tx) are coincident, such that E0ods(POIo(t)x) = E0pds(POIo(tx),tx) = E0pds(POIp) = #constant. Note that A formal derivation is needed (or just refer to Lucas). /*+-----+ (RFp) basis /% (mathH) E0ods(POIo,t) = |Q(PART)|/Rpcs(POIo(t),t)^2 (endMath) /*+-----+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (charge/length^2). (ignoring electric permeability for Gaussian coordinates) /*+-----+ (RFo) basis From "E0odv(POIo,t) = E0pdv(POIo(t),t)" : /% (5)* E0odv(POIo,t) = Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } /*Subbing (5)* into (2) : /% E0ods(POIo,t) = |E0odv(POIo)| = | Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } | = |Q(PART)|/{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } Summarizing : (mathH) E0ods(POIo,t) = |Q(PART)|/{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } (endMath) /*Key point from "Scalar absolute values, [vector, matrix] norms - simplification of expressions" : - EIpdv(POIo(t),t) - is in the opposite direction to E0pdv(POIo(t),t) vector norms - if some terms are vectors - multiplicative/divisive : |product(x1,x2,x3,...)| = product(|x1|,|x2|,|x3|,...) - subtraction/ addition : |sum (x1,x2,x3,...)| ≠ ≠ product(|x1|,|x2|,|x3|,...) in general (although it may be true in some cases) /********************* >>>>>>>>> Figure "BTodv(POIo,t)" http://www.BillHowell.ca/ /media/bill/SWAPPER/Lucas - Universal Force/Images/Howell - Chapter 4 - BTodv(POIo,t) - cropped.png /********************* >>>>>>>>> BIodv(POIo,t) = BIodv(POIp(t),t) ≠ BIpdv(POIp) = 0 Induced magnetic fields DO arise for Points Of Interest (POIo) that are fixed in the observer reference frame (RFo), as there is a relative velocity between the POIo and the particle, and therefore between POIo and ETodv(POIp(t),t), which in turn is the sum of E0odv(POIp(t),t) and EIodv(POIp(t),t)] (superposition applies - linear system). There is a recurrent nature to the derivation of an expression, as EIodv(POIp(t),t) itself depends on changes in the total magnetic field. This is covered in Section III "Chapter 4 - Derivations for a POI fixed in the observer reference frame (RFo)". /********************* >>>>>>>>> BTodv(POIo,t) = BTodv(POIp(t),t) ≠ BTpdv(POIp) = BTpdv(POIo(t),t) = 0 ???? Beginning with : /% BTodv(POIo,t) = B0odv(POIo,t) + BIodv(POIo,t) /*where B0odv(POIo,t) - magnetic field external (currents, permanent mags) in (RFo) BIodv(POIo,t) - magnetic field induced by charge Q(PART), which moves in RFo but does NOT move in (RFp) Later : This is addressed in "Howell - math of Lucas Universal Force.ndf", and further below where Lenz'sLaw and the Thomas Barnes iterations are addressed. /********************* >>>>>>>>> Prediction of direction of field (B), given that the current I flows in the direction of the thum Run command to see $ eog "$d_images""Wikipedia - Right-hand rule for B from E.png" & Prediction of direction of field (B), given that the current I flows in the direction of the thumb https://en.wikipedia.org/wiki/Right-hand_rule In order to compare to Lucas's intermediate results, it is handy to have an expression BEFORE invoking of Lenz's Law, and before Lucas's results from Thomas Barnes iterations. NOTE : How does Lucas's comment affect what is below? ???"... only static E fields give rise to B ..."??? p67h0.3 Equation (4-13) "... Another significant aspect of this work is that the induced B X v field is not electrostatic in nature. According to Cullwick [8], Hooper [10], and Moon and Spencer [12], this means that the linear superposition principle as applied to electric fields does not hold for the B X v generated fields. Thus in electrodynamics one must explicitly keep track of both electrostatic fields and the induced fields. Using the basic equations of electrodynamics one must calculate explicitly the induced fields in order to obtain the total fields of the moving charged particle. (Note that the covariant form of electrodynamics based upon Maxwell's equations assumes that the superposition principle holds and does not treat electrostatic and induced fields separately in disagreement with the experimental results cited. [10]) ..." (I wonder if he meant "E X v"?) BUT : equation (4-6) Lorentz force has v X B equation (4-11a) is a linear superposition of E0 & EI!!! equation (4-13) IS a linear supoerposition of E fields for calculating BI!!! WRONG!!! - While the EIo calculation is iterative (previous sub-subsection), the resulting EIo result is NOT used to calculate BIo, only the E0o portion applies. ??????? ????????? WRONG ??? : Assuming : /% BTpdv(POIo(t),t) = B0pdv(POIo(t),t) + (BIpdv(POIo(t),t) = 0) 27Aug2019 no BIpdv?? /*where B0pdv(POIo(t),t) - magnetic field external (currents, permanent mags) in (RFo) BIpdv(POIo(t),t) - magnetic field induced by the moving charge Q(PART) in (RFo) For the Chapter 4 situation, there is no "external" magnetic field (i.e. from other sources, independent of the particle), and so B0 is a constant zero (assuming that this is the definition of B0). Induced magnetic fields BIodv(POIo,t) do arise for Points Of Interest (POIo) that are fixed in the observer reference frame (RFo), as there is a relative velocity between the POIo and the particle (and therefore between POIo and E0ocv(POIo(t),t)). Points Of Interest (POIp) that are fixed in the particle reference frame (RFp) are covered in the previous section II "Chapter 4 - Derivations for a POIp fixed in the particle reference frame (RFp)". Here there is NO induced magnetic field as there is NO relative velocity between the POIp and the charged particle (and therefore between POIp and E0pcv(POIp)). /%From "E0odv(POIo,t) = E0pdv(POIo(t),t)" : (1) E0odv(POIo,t) = E0pdv(POIo(t),t) = Q(PART)/Rpcv(POIo(t),t)^2 and EIpdv(POIo(t),t) = EIodv(POIo,t) /* ASSUMING that E0odv(POIo,t) & EIodv(POIo,t) are in the same direction!!! Note : Lenz's Law (Lucas p64h0.5 Eq (4-5), p70h0.9 Eq (4-31)) provides assurance that E0odv(POIo,t) & EIodv(POIo,t) are in the same direction ... ??? 19Dec2017 NO! They are in OPPOSITE directions!!! This is important for scalar measures. /%From "ETodv(POIo,t)= ((ETpdv(POIo(t),t)=E0pdv(POIp))=E0pdv(POIp))" : (1) E0pdv(POIo(t),t) = E0pdv(POIo(t),t) + EIpdv(POIo(t),t) = E0odv(POIo,t) = E0odv(POIo,t) + EIodv(POIo,t) /* /% (mathH)/* Generalized Ampere's Law : /% BIodv(POIo,t) = Vonv(PART)/c X EOpdv(POIo(t),t) (endMath) /*From Lucas p67h0.6 Eqn (4-13) : /% (mathH)/* (4-13) Generalized Ampere's Law : /% BTpdv(POIo(t),t) = Vonv(PART)/c X [E0pdv(POIo(t),t) + EIpdv(POIo(t),t)] (endMath) /* ("X" is the vector cross-product operator) : /*+-----+ (RFp) basis From "E0odv(POIo,t) = E0pdv(POIo(t),t)", subbing (1)* into (4-13)* : /%(1) BTpdv(POIo(t),t) = BTodv(POIo,t) = Vonv(PART)/c X [ E0pdv(POIo(t),t) + EIpdv(POIo(t),t) ] = Vonv(PART)/c X Q(PART)/Rpcv(POIo(t),t)^2 + Vonv(PART)/c X EIpdv(POIo(t),t) /*Key point!! Now, when using scalars, it must be kept in mine that : - EIpdv(POIo(t),t) - is in the opposite direction to E0pdv(POIo(t),t) - from "Scalar absolute values, [vector, matrix] norms - simplification of expressions" vector norms - if some terms are vectors - multiplicative/divisive : |product(x1,x2,x3,...)| = product(|x1|,|x2|,|x3|,...) - subtraction/ addition : |sum (x1,x2,x3,...)| ≠ ≠ product(|x1|,|x2|,|x3|,...) in general (although it may be true in some cases) Therefore, EIpds(POIo(t),t) must be multiplied by -1 when added to E0pds(POIo(t),t) !!! /% = Vons(PART)/c *Q(PART)/Rpcs(POIo(t),t)^2 *sin[Aθpd(Vonv,Rpch(POIo(t),t))]*Rodh(Vonv_X_Rpcv(POIo(t),t)) - Vons(PART)/c *EIpds(POIo(t),t) *sin[Aθpd(Vonv,Rpch(POIo(t),t))]*Rodh(Vonv_X_Rpcv(POIo(t),t)) = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t) ] where : (2) Aθpd(Vonv,Rpch(POIo(t),t)) is the Aθ (theta) angle between the Vonv(PART) & E vectors (3) Aφpd(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction, which IS the direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) (4) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Note that it is a pure RFo unit vector, constant with time Notes : Aφpd(Vonv_X_Rpcv(POIo(t),t)) is the direction perpendicular to Voch X Rpch(POIo(t),t) using the right-hand rule. /*But : as this doesn't change in the Chapter 4 derivations it isn't shown in the formulae I haven't properly accounted for the signs yet (right hand rule). Notice that, because Vonh is the same as the O0pch = O0och coordinate direction for both (RFo) and (RFp) (i.e. it is parallel to the particle motion in (RFo)), the angle of intersection is simply Aθpc(POIo(t),t). In other words : /%(5) Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) /*Therefore : /% (mathH) BTpdv(POIo(t),t) = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t)] (endMath) where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθoc(POIo) (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφoc(POIo) (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Aφpd(Vonv_X_Rpcv(POIo(t),t)) & Rodh(Vonv_X_Rpcv(POIo(t),t)) are CONSTANT for Chapter 4 /*which is equivalent to Lucas Eqn (4-15) CAUTION!!! : ??? Lucas p67h0.3 - "B X v" field is not electrostatic in nature : what does this mean? Is this an error, should it be "E X v"???? /*+--+ Limit checks : Dimensional consistency, noting that unit vectors are taken as dimensionless : /% (6) BTpdv(POIo(t),t) = BTodv(POIo,t) = Vons(PART)/c *sin(Aθpc(POIo(t),t)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART)/Rpcs(POIo(t),t)^2 + EIpds(POIo(t),t) ] = (length/time)/(length/time) *(dimensionless) *(dimensionless) *[ (charge) /(length^2) + (charge/length^2) ] = (charge/length^2) /*OK as all units reduce to (charge/length^2) (ignoring permittivity & permeability for Gaussian units) /*+-----+ (RFo) basis Starting with the expression above in (RFp) coordinates, and noting that BTodv(POIo,t) = BTpdv(POIo(t),t) : /% (6) BTpdv(POIo(t),t) = BTodv(POIo,t) = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t) ] /*From "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)" : /% (2) Rpcs(POIo(t),t)) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) (5) sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /*Finally, noting that, as RFo and RFp are aligned, but with offset along the vov direction : (7) Rodh(Vonv_X_Rpcv(POIo(t),t)) = Rodh(Vonv_X_Rpcv(POIo)) NOTE: EIpdv(POIo(t),t) = EIodv(POIo,t) ASSUMING that E0odv(POIo,t) & EIodv(POIo,t) are in the same direction!!! Note : Lenz's Law (Lucas p64h0.5 Eq (4-5), p70h0.9 Eq (4-31)) provides assurance that E0odv(POIo,t) & EIodv(POIo,t) are in the same direction ... ??? /*Subbing (2)*, (6)*, and (7) into (6) : /%(8) BTpdv(POIo(t),t) = BTodv(POIo,t) = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART)/Rpcs(POIo(t),t)^2 - EIods(POIo(t),t) ] = Vons(PART)/c *Rodh(Vonv_X_Rpcv(POIo(t),t)) *sin(Aθpc(POIo(t),t)) *[ Q(PART) /Rpcs(POIo(t),t)^2 - EIods(POIo(t),t) ] = Vons(PART)/c *Rodh(Vonv_X_Rpcv(POIo(t),t)) * Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) *[ Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^2 - EIods(POIo(t),t) ] = Vons(PART)/c *Rocs(POIo)*sin(Aθoc(POIo)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) - EIods(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ] /*What is the geometric interpretation of ? : /% Rocs(POIo) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) = Rocs(POIo)/Rpcs(POIo(t),t)^3 Rocs(POIo) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) = Rocs(POIo)/Rpcs(POIo(t),t) /*Hmm... should be able to do something with this. Repeating the result : /% (mathH) BTodv(POIo,t) = Vons(PART)/c*Rocs(POIo)*sin(Aθoc(POIo))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) - EIods(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ] (endMath) where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) /*+--+ LIMIT CHECKS : Dimensional consistency, noting that unit vectors are taken as dimensionless : /% (9) BTpdv(POIo(t),t) = BTodv(POIo,t) = Vons(PART)/c *Rocs(POIo)*sin(Aθoc(POIo)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) + EIods(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ] = (length/time)/(length/time) *(length) *(dimensionless) *(dimensionless) *[ (charge) /(length^3) + (charge/length^2)/(length) ] = (length) *[ (charge/length^3) + (charge/length^3) ] = (charge/length^2) /*OK as all units reduce to (charge/length^2) (ignoring permittivity & permeability for Gaussian units) /********************* >>>>>>>>> EIodv(POIo,t) = EIodv(POIp(t),t) ≠ EIpdv(POIp) = 0 Again, the INDUCED electric field arises from CHANGES in the magnetic field at a point. In the observer reference frame RFo, the electric field at POIo = POIp(t) (that is, POIp at time t) DOES change, and gives rise to an induced field (which is derived in a later section). Later : This is addressed in "Howell - math of Lucas Universal Force.ndf", and further below where Lenz's Law and the Thomas Barnes iterations are addressed. /********************* >>>>>>>>> ETodv(POIo,t) = ETodv(POIp(t),t) = E0odv(POIp(t),t) + EIodv(POIp(t),t) ≠ ETpdv(POIp) = 0 Later : This is addressed in "Howell - math of Lucas Universal Force.ndf", and further below where Lenz'sLaw and the Thomas Barnes iterations are addressed. /% (mathH) EIods(POIo,t) = EIpds(POIo(t),t) ≠ EIpds(POIp) = 0 (endMath) /*For the "Basic math & calculus", [EIodv(POIo,t) = EIpdv(POIo(t),t), EIods(POIo(t),t) = EIpds(POIo(t),t)] (and their derivatives later) are being left "as is" in the expression. In later sections, Lenz's Induction Law and Thomas Barnes iterations will yield closed expressions for these variables. Key point from "Scalar absolute values, [vector, matrix] norms - simplification of expressions" : - EIpdv(POIo(t),t) - is in the opposite direction to E0pdv(POIo(t),t) vector norms - if some terms are vectors - multiplicative/divisive : |product(x1,x2,x3,...)| = product(|x1|,|x2|,|x3|,...) - subtraction/ addition : |sum (x1,x2,x3,...)| ≠ ≠ product(|x1|,|x2|,|x3|,...) in general (although it may be true in some cases) /********************* >>>>>>>>> ETodv(POIo,t) = ETpdv(POIo(t),t) Following Lucas p67h0.6 Eqn (4-13) & (4-41) : /% (1)* ((ETpdv(POIo(t),t)=E0pdv(POIp))=E0pdv(POIp)) = E0pdv(POIo(t),t) + EIpdv(POIo(t),t) /*From "" : Following Lucas p67h0.6 Eqn (4-13) & (4-41) : /% (2)* = ETodv(POIo,t) = E0odv(POIo,t) + EIodv(POIo,t) /********************* /%>>>>>>>>> ETods(POIo,t) = ((ETpds(POIo(t),t)=E0pds(POIp))=E0pds(POIp)) (mathH)/* Following Lucas p67h0.6 Eqn (4-13) & (4-41) /% E0pdv(POIo(t),t) + EIpdv(POIo(t),t) = ETodv(POIo,t) = E0odv(POIo,t) + EIodv(POIo,t) (endMath) /*For scalar calculations, must keep in mind Lenz's Law (induced is opposite direction of "inducer") Key point!! Now, when using scalars, it must be kept in mine that : - EIpdv(POIo(t),t) - is in the opposite direction to E0pdv(POIo(t),t) - from "Scalar absolute values, [vector, matrix] norms - simplification of expressions" 19Dec2017 OK - I get it now after coming back after ~1 year. This is required for the SCALAR equations! vector norms - if some terms are vectors - multiplicative/divisive : |product(x1,x2,x3,...)| = product(|x1|,|x2|,|x3|,...) - subtraction/ addition : |sum (x1,x2,x3,...)| ≠ ≠ product(|x1|,|x2|,|x3|,...) in general (although it may be true in some cases) Therefore : /% (1)* ETpds(POIo(t),t) = E0pds(POIo(t),t) - EIpds(POIo(t),t) (2)* = ETods(POIo,t) = E0ods(POIo,t) - EIods(POIo,t) /*+-----+ (RFp) basis From "E0ods(POIo,t) = E0pds(POIo(t),t)" : /% (mathH) E0pds(POIo(t),t) = Q(PART)/Rpcs(POIp)^2 (endMath) /*Subbing (1)* into (1) : ETpds(POIo(t),t) = E0pds(POIo(t),t) - EIpds(POIo(t),t) = Q(PART)/Rpcs(POIp(t),t)^2 - EIpds(POIo(t),t) (mathH)/* seems wrong???? /% ETpds(POIo(t),t) = Q(PART)/Rpcs(POIp(t),t)^2 - EIpds(POIo(t),t) (endMath) /*+-----+ (RFo) basis /%From "E0ods(POIo,t) = E0pds(POIo(t),t)" : (2)** E0ods(POIo) = |Q(PART)|/{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } Subbing (2)** into (2)* (2)* ETods(POIo(t),t) = E0ods(POIo,t) - EIods(POIo(t),t) So : (4) ETods(POIo(t),t) = |Q(PART)|/{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } - EIods(POIo(t),t) /***************************************** >>>>>> Derivatives /********************* >>>>>>>>> Figure "Calculus for RFo, using POIo=POIp(t)" http://www.BillHowell.ca/ /media/bill/SWAPPER/Lucas - Universal Force/Images/Howell - Chapter 4 - POIp calculus - cropped.png Reminders for Chapter 4 : Vonv(PART) = constant [Particle, observer] frames of reference (RFp) & (RFo) are IDENTICAL [scale, rotation, etc] at time t=0, apart from the given that the particle's reference frame (RFp) moves with velocity v relative to RFo. Reminders for (POIp) - Point Of Interest that is FIXED in the particle reference frame (RFp) : [Rpc,Opc,Ppc] and their derived concepts, are NOT functions of time, i.e. (POIp). Their derivatives with respect to time are zero. [Roc,Ooc,Poc], their related concepts, and their derivatives are all functions of time, i.e. (POIp(t),t) OBJECTIVES : Where convenient, express variables as functions of FIXED variables : (POIp) -> -Vodv(PART),t,Rpcv(POIp),Aθpc(POIp),Aφpc(POIp) (POIo) -> Vodv(PART),t,Rocv(POIo),Aθoc(POIo),Aφoc(POIo) /********************* >>>>>>>>> ∂[∂(t): Rocv(POIo)], ∂[∂(t): Aθoc(POIo)], ∂[∂(t): Aφoc(POIo)] = 0 As per "Rocv(POIo), Aθoc(POIo), Aφoc(POIo)" above, Roc and Ooc are constants for a given POIo, so their derivatives are zero : (mathH) ∂[∂(t): Rocv(POIo)] = 0 (endMath) (mathH) ∂[∂(t): Aθoc(POIo)] = 0 (endMath) (mathH) ∂[∂(t): Aφoc(POIo)] = 0 (endMath) /********************* >>>>>>>>> ∂[∂(t): Rocv(POIp(t),t) ] From Galilaean invariance rov - Vonv(PART)*t = Rpcv(POIp) = constant Also, for any POI in RFo (other than the particle itself), rov is a constant (fixed position for Chapter 4 - there are no other "pieces" moving with respect to RFo. Therefore /% ∂[∂(t): Rocv(POIp(t),t)] = ∂[∂(t): Rpcv(POIp) + Vonv(PART)*t] = ∂[∂(t): Rpcv(POIp)] + ∂[∂(t): Vonv(PART)*t] = 0 + Vonv(PART) (mathH) ∂[∂(t): Rocv(POIp(t),t) = Vonv(PART)] (endMath) /*This is expected. /*+-----+ Limit checks : Dimensional consistency - OK, as all terms reduce to (length/time). /********************* /%>>>>>>>>> ∂[∂(t): Rocs(POIp(t),t)] /*???Check this - improper derivative (use vector approach!!) /% Rocs(POIp(t),t) = |Rocv(POIp(t),t)| See "∂[∂(t): Rocv(POIp(t),t)] " above : ∂[∂(t): Rocs(POIp(t),t)] = ∂[∂(t): |Rocv(POIp(t),t)|] /*Kahan formulation : d||z|| = u_T dotPRod dz / ||z|| where u_T is the linear functional dual to z wrt ||...|| HERE I INTERPRET : u_T = Rocv(POIp(t),t)], so : /% ∂[∂(t): Rocs(POIp(t),t)] = ∂[∂(t): |Roc(POIp(t),t)|] = Rocv(POIp(t),t) dotPRod ∂[∂(t): Rocv(POIp(t),t)] / |Rocv(POIp(t),t)| from Eq (1) in "∂[∂(t): Rocv(POIp(t),t)]" above : ∂[∂(t): Rocv(POIp(t),t)] = Vonv(PART) = Rocv(POIp(t),t) dotPRod Vonv(PART) / |Rocv(POIp(t),t)| = |Rocv(POIp(t),t)|*cosAθoc(POIp(t),t)*|Vonv(PART)| / |Rocv(POIp(t),t)| = Vons(PART) *cosAθoc(POIp(t),t) (mathH) ∂[∂(t): Rocs(POIp(t),t)] = Vons(PART)*cosAθoc(POIp(t),t) (endMath) /*Want in terms of variables of (POIp) : /*+-----+ CONFIRMATION : Show Figure "∂[∂(t): Rocs(POIp(t),t)] explained" /%(2) |Rocv(POIp(t),t)| = Rocs(POIp(t),t) = { [(Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))]^2 + [Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]^2 }^(1/2) (3) ∂[∂(t): Rocs(POIp(t),t)] = ∂[∂(t): |Rocv(POIp(t),t)|] = 1/2*{ [(Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))]^2 + [Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]^2 }^(-1/2) *∂[∂(t): {(Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))]^2 + [Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]^2] (4) ∂[∂(t): [(Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))]^2 + [Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]^2] = ∂[∂(t): [(Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))]^2] + ∂[∂(t): Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]^2 ] But for all t, the following are constants, as Vodv(POI) is parallel to the coordinate axis [Rθ0och, Rθ0pch] : Rpcs(POIp)*sin(Aθpc(POIp)) = Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t)) = constant over t for any (POIo), (POIp) therefore ∂[∂(t): [(Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))]^2 + [Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]^2] = ∂[∂(t): [Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]^2] = 2 *[Rocs(POIp(t),t)*cosAθoc(POIp(t),t)] *∂[∂(t): Rocs(POIp(t),t)*cosAθoc(POIp(t),t)] Where from "∂[∂(t): Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]" below : ∂[∂(t): Rocs(POIp(t),t)*cosAθoc(POIp(t),t)] = Vons(PART) therefore (5) ∂[∂(t): [(Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))]^2 + [Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]^2] = 2*Vons(PART)*Rocs(POIp(t),t)*cosAθoc(POIp(t),t) Repeating (3) ∂[∂(t): Rocs(POIp(t),t)] = ∂[∂(t): |Rocv(POIp(t),t)|] = 1/2*{ [(Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))]^2 + [Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]^2 }^(-1/2) *∂[∂(t): {(Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))]^2 + [Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]^2] Substituting (5) into (3) : = 1/2*{ [(Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))]^2 + [Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]^2 }^(-1/2) *2*Vons(PART) * Rocs(POIp(t),t)*cosAθoc(POIp(t),t) = Vons(PART) * Rocs(POIp(t),t)*cosAθoc(POIp(t),t) *{ [(Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))]^2 + [Rocs(POIp(t),t)*cosAθoc(POIp(t),t)]^2 }^(-1/2) = Vons(PART) * Rocs(POIp(t),t)*cosAθoc(POIp(t),t) /Rocs(POIp(t),t)*{sin(Aθoc(POIp(t),t))^2 + cosAθoc(POIp(t),t) ^2 }^(-1/2) But 1 = { sin(Aθoc(POIp(t),t))^2 + cosAθoc(POIp(t),t)^2 }^(-1/2) therefore : (6) ∂[∂(t): Rocs(POIp(t),t)] = Vons(PART)*cosAθoc(POIp(t),t) /*This is the same as (1) above, confirming that result. /*+-----+ OBJECTIVES : Where convenient, express variables as functions of FIXED variables : (POIp) -> -Vodv(PART),t,Rpcv(POIp),Aθpc(POIp),Aφpc(POIp) (POIo) -> Vodv(PART),t,Rocv(POIo),Aθoc(POIo),Aφoc(POIo) /*From "Relating [Rocv,Rocs,ROPI2ocs ,rO0ocs,sin(Aθoc),cos(Aθoc)]@(POIp(t),t) to [...]@(POIp)" : /% (2) Rocs(POIp(t),t)) = { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIp)*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) so : Rocs(POIp(t),t) = { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cosAθpc(POIp)*Vons(PART)*t + (Vons(PART)*t)^2 }^( 1/2) and : ∂[∂(t): Rocs(POIp(t),t)] = (1/2)*{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cosAθpc(POIp)*Vons(PART)*t + (Vons(PART)*t)^2 }^(-1/2) * ( 0 + 2*Rpcs(POIp)*cosAθpc(POIp)*Vons(PART) + 2*(Vons(PART)*t)*Vons(PART) } = (1/2)*{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cosAθpc(POIp)*Vons(PART)*t + (Vons(PART)*t)^2 }^(-1/2) * { 0 + 2*Rpcs(POIp)*cosAθpc(POIp)*Vons(PART) + 2 *(Vons(PART)*t)*Vons(PART) } = { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cosAθpc(POIp)*Vons(PART)*t + (Vons(PART)*t)^2 }^(-1/2) *{ Rpcs(POIp)*cosAθpc(POIp)*Vons(PART) + Vons(PART)^2*t } then : (mathH) ∂[∂(t): Rocs(POIp(t),t)] = {Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + Vons(PART)^2*t} / {Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + (Vons(PART)*t)^2 }^(1/2) (endMath) /*+-----+ Limit checks : Dimensional consistency - OK, as all terms reduce to (length/time). As t -> +- infinity : /% (7) ∂[∂(t): Rocs(POIp(t),t)] = ( alpha(POIp) + Vons(PART)^2*t } /{ Rpcs(POIp)^2 + 2*alpha(POIp)*t + (Vons(PART) *t)^2 }^(1/2) tends to : => Vons(PART)^2*t / { (Vons(PART)*t)^2 }^(1/2) = +- Vons(PART) which is expected. Limit check : As t -> +- 0 : (7) ∂[∂(t): Rocs(POIp(t),t)] = ( alpha(POIp) + Vons(PART)^2*t } /{ Rpcs(POIp)^2 + 2*alpha(POIp)*t + (Vons(PART) *t)^2 }^(1/2) = ( alpha(POIp) + 0 } /{ Rpcs(POIp)^2 + 0 + 0 }^(1/2) = alpha(POIp) / Rpcs(POIp) where alpha(POIp) = Rpcs(POIp)*cosAθpc(POIp)*Vons(PART) so : ∂[∂(t): Rocs(POIp(t),t)] = Rpcs(POIp)*cosAθpc(POIp)*Vons(PART) / Rpcs(POIp) = cosAθpc(POIp)*Vons(PART) /*Harder to tell this is OK... Limit check : /%As cosAθoc(POIp(t),t) -> 0 : Vons(PART)*t = -cosAθpc(POIp)*Rpcs(POIp) /********************* /%>>>>>>>>> ∂[∂(t): sin(Aθoc(POIp(t),t))] /*For Chapter 4, v = constant, [Particle, observer] frames of reference are IDENTICAL [scale, rotation, etc] at time t=0, apart from the given that the particle's reference frame (RFp) moves with velocity v relative to RFo. See "Howell - math of Lucas Universal Force.ndf" (4-17) 'Sperical coordinate transforms ' Figure "Chapter 4 reference frames" /% Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t)) = Rpcs(POIp)*sin(Aθpc(POIp)) sin(Aθoc(POIp(t),t)) = Rpcs(POIp)/Rocs(POIp(t),t)*sin(Aθpc(POIp)) but [Rpcs(POIp),sin(Aθpc(POIp))] are constants, therefore ∂[∂(t): sin(Aθoc(POIp(t),t))] = Rpcs(POIp)*sin(Aθpc(POIp)) * ∂[∂(t): 1/Rocs(POIp(t),t)] from "∂[∂(t): Rocs(POIp(t),t)] = ∂[∂(t): |Rocv(POIp(t),t)|]" : (1) ∂[∂(t): Rocs(POIp(t),t)] = Vons(PART)*cosAθoc(POIp(t),t) therefore (1) ∂[∂(t): sin(Aθoc(POIp(t),t))] = Rpcs(POIp)*sin(Aθpc(POIp))*(-1)/Rocs(POIp(t),t)^2*Vons(PART)*cosAθoc(POIp(t),t) = -Rpcs(POIp)*sin(Aθpc(POIp))*Vons(PART) *cosAθoc(POIp(t),t)/Rocs(POIp(t),t)^2 I want to express this all in the particle reference frame (RFp) -> (POIp), Vons(PART), t Again : Rpcs(POIp)*sin(Aθpc(POIp)) = Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t)) therefore (2) cosAθoc(POIp(t),t)/Rocs(POIp(t),t)^2 = cosAθoc(POIp(t),t)* [ sin(Aθoc(POIp(t),t))/Rpcs(POIp)/sin(Aθpc(POIp)) ]^2 /*From "Relating [Rocv,Rocs,ROPI2ocs ,rO0ocs,sin(Aθoc),cos(Aθoc)]@(POIp(t),t) to [...]@(POIp)" : /% (5) sin(Aθoc(POIp(t),t)) = Rpcs(POIp)*sin(Aθpc(POIp)) /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /*From "Relating [Rocv,Rocs,ROPI2ocs ,rO0ocs,sin(Aθoc),cos(Aθoc)]@(POIp(t),t) to [...]@(POIp)" : : /% (6) cos(Aθoc(POIp(t),t)) = [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] /{Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing into (2) : (3) cosAθoc(POIp(t),t)*sin(Aθoc(POIp(t),t))^2 = [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) * [ Rpcs(POIp)*sin(Aθpc(POIp)) ]^2 / { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } = [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] * [ Rpcs(POIp)*sin(Aθpc(POIp)) ]^2 / { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) Subbbing end result of (3) into (2) : (4) 2) cosAθoc(POIp(t),t)/Rocs(POIp(t),t)^2 = cosAθoc(POIp(t),t)* [ sin(Aθoc(POIp(t),t))/Rpcs(POIp)/sin(Aθpc(POIp)) ]^2 = [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] * [ Rpcs(POIp)*sin(Aθpc(POIp)) ]^2 / { [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 }^(3/2) / [ Rpcs(POIp)*sin(Aθpc(POIp)) ]^2 = [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / { [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 }^(3/2) Subbbing end result of (4) into (1) : (1) ∂[∂(t): sin(Aθoc(POIp(t),t))] = -Rpcs(POIp)*sin(Aθpc(POIp))*Vons(PART) *cosAθoc(POIp(t),t)/Rocs(POIp(t),t)^2 = -Rpcs(POIp)*sin(Aθpc(POIp)) *Vons(PART) [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / { [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 }^(3/2) (mathH) ∂[∂(t): sin(Aθoc(POIp(t),t))] = -Rpcs(POIp)*sin(Aθpc(POIp)) * Vons(PART) * [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / { [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 }^(3/2) (endMath) /*+-----+ ALTERNATIVE derivation /% (1) ∂[∂(t): sin(Aθoc(POIp(t),t))] = -Rpcs(POIp)*sin(Aθpc(POIp))*Vons(PART) *cosAθoc(POIp(t),t)/Rocs(POIp(t),t)^2 /*From "Relating [Rocv,Rocs,ROPI2ocs ,rO0ocs,sin(Aθoc),cos(Aθoc)]@(POIp(t),t) to [...]@(POIp)" : /% (2) Rocs(POIp(t),t)) = |Rpcv(POIp) + Vonv(PART)*t| = { [Rpcs(POIp)*cos(Aθpc(POIp)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 }^(1/2) /*From "Relating [Rocv,Rocs,ROPI2ocs ,rO0ocs,sin(Aθoc),cos(Aθoc)]@(POIp(t),t) to [...]@(POIp)" : : /% (6) cos(Aθoc(POIp(t),t)) = [ Rpcs(POIp)*cos(Aθpc(POIp)) + Vons(PART)*t ] / { [Rpcs(POIp)*cos(Aθpc(POIp)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 }^(1/2) so : ∂[∂(t): sin(Aθoc(POIp(t),t))] = -Rpcs(POIp)*sin(Aθpc(POIp)) *Vons(PART) *cosAθoc(POIp(t),t)/Rocs(POIp(t),t)^2 = -Rpcs(POIp)*sin(Aθpc(POIp)) *Vons(PART) * [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / { [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 }^(1/2) / { [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 } therefore : (6) ∂[∂(t): sin(Aθoc(POIp(t),t))] = -Rpcs(POIp)*sin(Aθpc(POIp)) * Vons(PART) * [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / { [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 }^(3/2) /*Which is the same as (5) above. Equation (5) = Equation (6), so at least this checks - but these aren't really independent. /*+-----+ Limit checks : Dimensional consistency - OK, as all terms reduce to (time)^(-1). As (6) = (5), take this as a confirmation of sorts. /%As cos(Aθpc(POIo(t),t)) -> 0 (therefore sin(Aθpc(POIp)) -> +- 1) /********************* /%>>>>>>>>> ∂[∂(t): cosAθoc(POIp(t),t)] /*Reminders for Chapter 4 : Vonv(PART) = constant [Particle, observer] frames of reference (RFp) & (RFo) are IDENTICAL [scale, rotation, etc] at time t=0, apart from the given that the particle's reference frame (RFp) moves with velocity Vonv(PART) relative to RFo. Reminders for (POIp) - Point Of Interest that is FIXED in the particle reference frame (RFp) : [Rpc,Opc,Ppc] and their derived concepts, are NOT functions of time, i.e. (POIp). Their derivatives with respect to time are zero. [Roc,Ooc,Poc], their derived concepts, and their derivatives are all functions of time, i.e. (POIp(t),t) OBJECTIVES : Where convenient, express variables as functions of FIXED variables : (POIp) -> -Vodv(PART),t,Rpcv(POIp),Aθpc(POIp),Aφpc(POIp) (POIo) -> Vodv(PART),t,Rocv(POIo),Aθoc(POIo),Aφoc(POIo) /*+-----+ Cosine expression /*From "Relating [Rocv,Rocs,ROPI2ocs ,rO0ocs,sin(Aθoc),cos(Aθoc)]@(POIp(t),t) to [...]@(POIp)" : Distance of RO0ocs(POIp(t),t) from the (RFo) origin in RO0och direction (i.e. along L(PART)) : /% (4) Rθ0ocs(POIp(t),t) = Rocs(POIp(t),t)*cos(Aθoc(POIp(t),t)) = Rpcs(POIp)*cos(Aθpc(POIp)) + Vons(PART)*t therefore : cos(Aθoc(POIp(t),t)) = [Rpcs(POIp)*cos(Aθpc(POIp)) + Vons(PART)*t ] / Rocs(POIp(t),t) /*From "Relating [Rocv,Rocs,ROPI2ocs ,rO0ocs,sin(Aθoc),cos(Aθoc)]@(POIp(t),t) to [...]@(POIp)" : /% (2) Rocs(POIp(t),t)) = { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) therefore : cos(Aθoc(POIp(t),t)) = [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / Rocs(POIp(t),t) = [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /*+-----+ Expression for ∂[∂(t): cosAθoc(POIp(t),t)] ∂[∂(t): cosAθoc(POIp(t),t)] = ∂[∂(t): [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ] = ∂[∂(t): [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] ] /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) + [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] *∂[∂(t):{Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-1/2)] = Vons(PART) /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) + [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] *{(-1/2)/{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) *∂[∂(t): { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } } = Vons(PART) /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) - [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] *{ /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) * (1/2)*{ 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + 2*[Vons(PART)*t] *∂[∂(t): [Vons(PART)*t] ] } } = Vons(PART) /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) - *[ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] *{1/{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) * { Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + [Vons(PART)^2*t] } } = Vons(PART) / { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) - [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] * { Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + [Vons(PART)^2*t] } / { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) = Vons(PART) /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) - *[ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] *{ + Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + [Vons(PART)^2*t] } /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) /*Leaving it at that : /% (mathH) ∂[∂(t): cosAθoc(POIp(t),t)] = Vons(PART) / {Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART) *t]^2}^(1/2) - { Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART) *t} *{ + Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + [Vons(PART)^2*t] } /{Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART) *t]^2}^(3/2) (endMath) /*+-----+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (time)^(-1). Rewriting in terms of Rocs(POIp(t),t) : Limit check /********************* /%>>>>>>>>> ∂[∂(t): Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))] /********************* >>>>>>>>> Figure "Calculus for a POIp", it is clear BY INSPECTION that : (mathH) ∂[∂(t): Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))] = ∂[∂(t): ROPI2ods(POIp(t),t)] = 0 (endMath) /%This makes sense, as from the Figure, Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t)) is a constant because the endpoint of Rocv is always on Line(POIo), parallel to Line(PART), the trajectory of the center of the particle. Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t)) = Rpcs(POIp)*sin(Aθpc(POIp)) /*+-----+ ALTERNATIVE DERIVATION : This also acts as a check on [consistency, correctness] of formulae. /%(2) ∂[∂(t): Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))] = ∂[∂(t): Rocs(POIp(t),t)]*sin(Aθoc(POIp(t),t)) + Rocs(POIp(t),t)*∂[∂(t): sin(Aθoc(POIp(t),t))] FIRST TERM from "∂[∂(t): Rocs(POIp(t),t)] = ∂[∂(t): |Rocv(POIp(t),t)|]" : (7) ∂[∂(t): Rocs(POIp(t),t)] = ( Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + Vons(PART)^2*t } /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + (Vons(PART) *t)^2 }^(1/2) from "Relating [Roc,sin(Aθoc(POIo)),... ] @(POIp(t),t) to [Rpcv,sin(Aθpc(POIp)),...] @(POIp)" (4) sin(Aθoc(POIp(t),t)) = Rpcs(POIp)*sin(Aθpc(POIp)) /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) So : (3) ∂[∂(t): Rocs(POIp(t),t)]*sin(Aθoc(POIp(t),t)) = ( Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + Vons(PART)^2*t } /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + (Vons(PART) *t)^2 }^(1/2) * Rpcs(POIp)*sin(Aθpc(POIp)) /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART) *t]^2 }^(1/2) = ( Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + Vons(PART)^2*t } * Rpcs(POIp)*sin(Aθpc(POIp)) /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART) *t]^2 } SECOND TERM from "∂[∂(t): sin(Aθoc(POIp(t),t))]" : (5) ∂[∂(t): sin(Aθoc(POIp(t),t))] = -Rpcs(POIp)*sin(Aθpc(POIp)) *Vons(PART) * [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / { [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 }^(3/2) /*from "Relating [Rocv,Rocs,ROPI2ocs ,rO0ocs,sin(Aθoc),cos(Aθoc)]@(POIp(t),t) to [...]@(POIp)" /% (2) Rocs(POIp(t),t)) = { Rpcs(POIp)^2 + 2*Rpcs(POIp) *cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) therefore : (4) Rocs(POIp(t),t)*∂[∂(t): sin(Aθoc(POIp(t),t))] = { Rpcs(POIp)^2 + 2*Rpcs(POIp) *cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) * -Rpcs(POIp)*sin(Aθpc(POIp)) *Vons(PART) * [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / { [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 }^(3/2) = -Rpcs(POIp)*sin(Aθpc(POIp)) *Vons(PART) * [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / { Rpcs(POIp)^2 + 2*Rpcs(POIp) *cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } PUTTING IT ALL TOGETHER : Subbing (3)&(4) into (2) : (2) ∂[∂(t): Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))] = ∂[∂(t): Rocs(POIp(t),t)]*sin(Aθoc(POIp(t),t)) + Rocs(POIp(t),t)*∂[∂(t): sin(Aθoc(POIp(t),t))] = { ( Rpcs(POIp)*cos(Aθpc(POIo(t),t)) *Vons(PART) + Vons(PART)^2*t } * Rpcs(POIp)*sin(Aθpc(POIp)) / { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t)) *Vons(PART)*t + [Vons(PART) *t]^2 } } +{ -Rpcs(POIp)*sin(Aθpc(POIp)) *Vons(PART) * [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t)) *Vons(PART)*t + [Vons(PART) *t]^2 } } = { ( Rpcs(POIp)^2*sin(Aθpc(POIp))*cos(Aθpc(POIo(t),t)) *Vons(PART) + Rpcs(POIp) *sin(Aθpc(POIp)) *Vons(PART)^2*t } / { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t)) *Vons(PART)*t + [Vons(PART) *t]^2 } } +{ -Rpcs(POIp) *sin(Aθpc(POIp)) *Vons(PART)*Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + -Rpcs(POIp) *sin(Aθpc(POIp)) *Vons(PART)*Vons(PART)*t ] / { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART) *t]^2 } } =[{ (Rpcs(POIp)^2*sin(Aθpc(POIp))*cos(Aθpc(POIo(t),t))*Vons(PART) + Rpcs(POIp) *sin(Aθpc(POIp)) *Vons(PART)^2*t } } -{ Rpcs(POIp)^2*sin(Aθpc(POIp))*cos(Aθpc(POIo(t),t))*Vons(PART) + Rpcs(POIp) *sin(Aθpc(POIp)) *Vons(PART)^2*t ] } ] /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t)) *Vons(PART)*t + [Vons(PART) *t]^2 } = 0 /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t)) *Vons(PART)*t + [Vons(PART) *t]^2 } = 0 (5) ∂[∂(t): Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))] = 0 /*As expected. /*+-----+ Limit checks : Dimensional consistency - OK in intermediate equations, as all terms reduce to (length/time). The agreement between the [geometric, algebraic] approaches above provides a small but essential degree of confirmation of the result. /********************* /%>>>>>>>>> ∂[∂(t): Rocs(POIp(t),t)*cosAθoc(POIp(t),t)] /*From the Figure "Calculus for a POIp" : /% (mathH) ∂[∂(t): Rocs(POIp(t),t)*cosAθoc(POIp(t),t)] = ∂[∂(t): Rθ0ocs(POIp(t),t)] = Vons(PART) (endMath) /*This makes sense, as from the Figure, Rocs(POIp(t),t)*cosAθoc(POIp(t),t)increases directly with Vons(PART). /*+-----+ ALTERNATIVE DERIVATION This also acts as a check on [consistency, correctness] of formulae. /%(2) ∂[∂(t): Rocs(POIp(t),t)*cosAθoc(POIp(t),t)] = ∂[∂(t): Rocs(POIp(t),t)]*cosAθoc(POIp(t),t) + Rocs(POIp(t),t)*∂[∂(t): cosAθoc(POIp(t),t)] /*+-----+ /%FIRST TERM : ∂[∂(t): Rocs(POIp(t),t)]*cosAθoc(POIp(t),t) from "∂[∂(t): Rocs(POIp(t),t)] = ∂[∂(t): |Rocv(POIp(t),t)|]" : (7) ∂[∂(t): Rocs(POIp(t),t)] = ( Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + Vons(PART)^2*t } /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + (Vons(PART) *t)^2 }^(1/2) /*from "Relating [Rocv,Rocs,ROPI2ocs ,rO0ocs,sin(Aθoc),cos(Aθoc)]@(POIp(t),t) to [...]@(POIp)" : /% (6) cos(Aθoc(POIp(t),t)) = [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] /{Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (7)&(6) into (2) : (3) ∂[∂(t): Rocs(POIp(t),t)]*cos(Aθoc(POIp(t),t)) = ( Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + Vons(PART)^2*t } /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + (Vons(PART) *t)^2 }^(1/2) *[ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) = { Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + Vons(PART)^2*t } *[ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } /*+-----+ /%SECOND TERM : Rocs(POIp(t),t)*∂[∂(t): cosAθoc(POIp(t),t)] /*from "Relating [Rocv,Rocs,ROPI2ocs ,rO0ocs,sin(Aθoc),cos(Aθoc)]@(POIp(t),t) to [...]@(POIp)" /% (2) Rocs(POIp(t),t)) = { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) from "∂[∂(t): cos(Aθoc(POIp(t),t))]" : (1) ∂[∂(t): cosAθoc(POIp(t),t)] = Vons(PART) /{Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) - [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] *{ Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + [Vons(PART)^2*t] } /{Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) therefore : (4) Rocs(POIp(t),t)*∂[∂(t): cosAθoc(POIp(t),t)] = {Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) *{ Vons(PART) /{Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) -[ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] *{ Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + [Vons(PART)^2*t] } /{Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) } = Vons(PART) -[ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] *{ Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + [Vons(PART)^2*t] } /{Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } /*+-----+ PUTTING IT ALL TOGETHER : Subbing (3)&(4) into (2) : /% (2) ∂[∂(t): Rocs(POIp(t),t)*cosAθoc(POIp(t),t)] = ∂[∂(t): Rocs(POIp(t),t)]*cosAθoc(POIp(t),t) + Rocs(POIp(t),t)*∂[∂(t): cosAθoc(POIp(t),t)] ={ ( Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + Vons(PART)^2*t ) * [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART) *t ] /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } } + { Vons(PART) - [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] *{ Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + [Vons(PART)^2*t] } /{Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } } = Vons(PART) + [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART) *t ] *( Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + Vons(PART)^2*t } /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } - [ Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART) *t ] *{ Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + [Vons(PART)^2*t] } /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 } = Vons(PART) (5) ∂[∂(t): Rocs(POIp(t),t)*cosAθoc(POIp(t),t)] = Vons(PART) /*As expected... Limit checks : Dimensional consistency - OK in result & intermediate equations, as all terms reduce to (length/time). The agreement between the [geometric, algebraic] approaches above provides a small but essential degree of confirmation of the result. endsection /********************* >>>>>>>>> Figure "Electrostatic field basics & calculus for a POIo" http://www.BillHowell.ca/ /media/bill/SWAPPER/Lucas - Universal Force/Images/Howell - Chapter 4 - Electrostatic field basics & calculus - cropped.png Differential vector geometry of ∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)] Looking at Figure "Electrostatic field basics & calculus for a POIo" : IMPORTANT!! I must apply the spherical coordinate derivative formulation to get the same result!!!!!!!! /*+--+ /********************* >>>>>>>>> DEFINITIONS /%1a) Aθpd(RDEpdh(POIo(t),∂(t))) is the direction perpendicular to Rpch(POIo(t),t), rotated in the positive Aθpn (or Aθon) direction (i.e at an angle of PI/2 from Rpch(POIo(t),t)). This gives an angle with respect to the (RFo) theta coordinate vector, Aθpc(POIo(t),t), of (PI/2). Aθpd(RDEpdh(POIo(t),∂(t))) defines a tanjent direction, and it is used below as a reference direction for chord that is important to derive the differential it is used below as ????????? Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 1b) RDEpdh(POIo(t),∂(t)) is a unit vector (dimensionless) in the direction of Aθpd(RDEpdh(POIo(t),∂(t))) 1c) RDEpds(POIo(t),∂(t)) = |RDEpdh(POIo(t),∂(t))| = 1 is the magnitude of the unit vector RDEpdh(POIo(t),∂(t)) 1d) E0DIFF_odv(POIo(t),∂(t)) = ∂[∂(t): E0odv(POIo,t)] is the vector DIFFerential change in E0odv(POIo,t)] 1e) E0DIFF_ods(POIo(t),∂(t)) = |∂[∂(t): E0odv(POIo,t)]| = E0DIFF_ods(POIo(t),∂(t)) is the magnitude of the DIFFerential change in E0odv(POIo,t)] 1f) Aθ_E0DIFFoda(POIo(t),∂(t)) is the direction of ∂[∂(t): E0odv(POIo,t)] with respect to E0odv(POIo,t), the latter being the same direction as Rpcv(POIo(t),t) 1g) Aθ_E0DIFFoca(POIo(t),∂(t)) is the direction of ∂[∂(t): E0odv(POIo,t)] with respect to E0odv(POIo,t), the latter being the same direction as Rpcv(POIo(t),t) : Aθ_E0DIFFoca(POIo(t),∂(t)) = Aθ_E0DIFFoca(POIo(t),∂(t)) + Aθpd(RDEpdh(POIo(t),∂(t))) + Aθpc(POIo(t),t) = Aθ_E0DIFFoca(POIo(t),∂(t)) + PI/2 + Aθpc(POIo(t),t) Looking at Figure "∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)]" : (1) ∂[∂(t): E0pdv(POIo(t),t)] = E0pds(POIo(t),t)*∂[∂(t): Aθpc(POIo(t),t)]*RDEpdh(POIo(t),∂(t)) + ∂[∂(t): |E0pdv(POIo(t),t)|] *Rpch(POIo(t),t) /*... ignoring differential changes in directions /%(2) ∂[∂(t): E0odv(POIo,t)] = E0ods(POIo,t)*∂[∂(t): Aθpc(POIo(t),t)]*RDEpdh(POIo(t),∂(t)) + ∂[∂(t): |E0odv(POIo,t)|] *Rpch(POIo(t),t) /*... ignoring differential changes in directions /********************* /%>>>>>>>>> ∂[∂(t): E0odv(POIo)] ≠ ∂[∂(t): E0pdv(POIp)] = 0 /*Reminders for Chapter 4 : Vonv(PART) = constant [Particle, observer] frames of reference (RFp) & (RFo) are IDENTICAL [scale, rotation, etc] at time t=0, apart from the given that the particle's reference frame (RFp) moves with velocity Vonv(PART) relative to RFo. Reminders for (POIo) - Point Of Interest that is FIXED in the observer reference frame (RFo) : [Roc,Ooc,Poc] and their derived concepts, are NOT functions of time, i.e. (POIp). Their derivatives with respect to time are zero. [Rpc,Opc,Ppc], their derived concepts, and their derivatives are all functions of time, i.e. (POIo(t),t) /*+-----+ (RFp) basis /%Looking at Figure "∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)]" : From "Differential geometry of ∂[∂(t): BTodv(POIo,t)] = ∂[∂(t): BTpdv(POIo(t),t)]" : ∂[∂(t): E0pdv(POIo(t),t)] = E0pds(POIo(t),t)*∂[∂(t): Aθpc(POIo(t),t)] *RDEpdh(POIo(t),∂(t)) + ∂[∂(t): E0pds(POIo(t),t)]*Rpch(POIo(t),t) ... ignoring differential change in direction for RDEpdh(POIo(t),∂(t)) & Rpch(POIo(t),t) /*+-----+ FIRST TERM /%(1) E0pds(POIo(t),t)*∂[∂(t): Aθpc(POIo(t),t)]*RDEpdh(POIo(t),∂(t)) where, as in "Differential vector geometry of ∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)]" : 1a) Aθpd(RDEpdh(POIo(t),∂(t))) is the direction perpendicular to Rpch(POIo(t),t), rotated in the positive Aθpn (or Aθon) direction (i.e at an angle of PI/2 from Rpch(POIo(t),t)). This gives an angle with respect to the (RFo) theta coordinate vector of (PI/2). Aθpd(RDEpdh(POIo(t),∂(t))) defines a tanjent direction, and it is used below as a reference direction for chord that is important to derive the differential it is used below as ????????? 1c) RDEpdh(POIo(t),∂(t)) is a unit vector in the direction of Aθpd(RDEpdh(POIo(t),∂(t))) From "E0ods(POIo,t) = E0pds(POIo(t),t)" : (1) E0pds(POIo(t),t) = Q(PART)/Rpcs(POIp)^2 From "∂[∂(t): Aθpc(POIo(t),t)]" : (5) d[∂(t): Aθpc(POIo(t),t)] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) Subbing (1)* & (5)* into (1) : (2) E0pds(POIo(t),t)*∂[∂(t): Aθpc(POIo(t),t)] *RDEpdh(POIo(t),∂(t)) = Q(PART) /Rpcs(POIp)^2 *Vons(PART)/Rpcs(POIo(t),t) *sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) = Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3*sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) /*+--+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (charge/length^2/time). (ignoring electric permeability for Gaussian coordinates) /*+-----+ SECOND TERM /%(3) ∂[∂(t): E0pds(POIo(t),t)]*Rpch(POIo(t),t) ... ignoring differential change in direction From "∂[∂(t): E0pds(POIo(t),t)] = ∂[∂(t): E0pds(POIo(t),t)]" : 4*) ∂[∂(t): E0pds(POIo(t),t)] = 2*Q(PART)*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIp)^3 Subbing (4)* into (3) : (4) ∂[∂(t): |E0pdv(POIo(t),t)|] = 2*Q(PART)*Vons(PART)/Rpcs(POIp)^3*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) /*+--+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (charge/length^2/time). (ignoring electric permeability for Gaussian coordinates) /*+-----+ COMBINING TERMS Subbing (2)&(4) into (1) : /% (1) ∂[∂(t): E0pdv(POIo(t),t)] = E0pds(POIo(t),t)*∂[∂(t): Aθpc(POIo(t),t)] *RDEpdh(POIo(t),∂(t)) + ∂[∂(t): |E0pdv(POIo(t),t)|] *Rpch(POIo(t),t) = Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3*sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*Q(PART)*Vons(PART)/Rpcs(POIp)^3*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) /*Therefore : /% ∂[∂(t): E0pdv(POIo(t),t)] = Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3*sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*Q(PART)*Vons(PART)/Rpcs(POIp)^3*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) /*+--+ /%Looking at Rpch(POIo(t),t), RDEpdh(POIo(t),∂(t)) By definition, Rpch(POIo(t),t) is in the direction of Rpcv(POIo(t),t), at angle Aθpc(POIo(t),t). /*From "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)" : /% (1) Rpcv(POIo(t),t) = Rocv(POIo) - Vonv(PART)*t From "Differential vector geometry of ∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)]" : 1a) Aθpd(RDEpdh(POIo(t),∂(t))) is the direction perpendicular to Rpch(POIo(t),t), rotated in the positive Aθpn (or Aθon) direction (i.e at an angle of PI/2 from Rpch(POIo(t),t)). This gives an angle with respect to the (RFo) theta coordinate vector, Aθpc(POIo(t),t), of (PI/2). Aθpd(RDEpdh(POIo(t),∂(t))) defines a tanjent direction, and it is used below as a reference direction for chord that is important to derive the differential it is used below as ????????? Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 1b) RDEpdh(POIo(t),∂(t)) is a unit vector (dimensionless) in the direction of Aθpd(RDEpdh(POIo(t),∂(t))) 1c) RDEpds(POIo(t),∂(t)) = |RDEpdh(POIo(t),∂(t))| = 1 is the magnitude of the unit vector RDEpdh(POIo(t),∂(t)) Summary : (mathH) ∂[∂(t): E0pdv(POIo(t),t)] = Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t)] (endMath) where : Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 /*+-----+ (RFo) basis : Where convenient, express variables as functions of FIXED variables : (POIp) -> -Vodv(PART),t,Rpcv(POIp),Aθpc(POIp),Aφpc(POIp) (POIo) -> Vodv(PART),t,Rocv(POIo),Aθoc(POIo),Aφoc(POIo) From "Relating [Rpcs,RO0pcs,ROPI2pcs,sin(Aθpc),cos(Aθpc)]@t to [Roc,AOo,APo] for (POIo)" : /% (2) Rpcs(POIo(t),t)) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) (5) sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) (6) cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) Subbing (2)*,(5)*(6)* into (6) : (6) ∂[∂(t): E0pdv(POIo(t),t)] = Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[ sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) ] where : Aθpd(Rpch(POIo(t),t)) = Aθpc(POIo(t),t) Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 = Q(PART)*Vons(PART) *[ RDEpdh(POIo(t),∂(t)) * Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) + Rpch(POIo(t),t) *2*[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^3 = Q(PART)*Vons(PART) *[ RDEpdh(POIo(t),∂(t)) * Rocs(POIo)*sin(Aθoc(POIo)) + Rpch(POIo(t),t) *2*[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^3 / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) = Q(PART)*Vons(PART) *[ RDEpdh(POIo(t),∂(t)) * Rocs(POIo)*sin(Aθoc(POIo)) + Rpch(POIo(t),t) *2*[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 (7) = Q(PART)*Vons(PART) *[ Rocs(POIo)*sin(Aθoc(POIo)) *RDEpdh(POIo(t),∂(t)) + 2*[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rpch(POIo(t),t) ] ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 /*+--+ Conversions of directions : From (6) : /% Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 /*As above, from "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)" : /% (5) sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) (6) cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) So Aθpc(POIo(t),t) can be expressed either as : (8) Aθpc(POIo(t),t) = arcsin [ Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ] or = arccos [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) ] Summarizing (7),(8),(9) : (mathH) ∂[∂(t): E0pdv(POIo(t),t)] = Q(PART)*Vons(PART) * {Rocs(POIo)*sin(Aθoc(POIo))*RDEpdh(POIo(t),∂(t)) + 2*[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rpch(POIo(t),t)] } / {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 (endMath) where : Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 Aθpc(POIo(t),t) = arcsin [ Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ] = arccos [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) ] /*+--+ LIMIT CHECKS : Dimensional consistency for magnitude - OK, as all units reduce to (charge/length^2/t) Dimensional consistency for direction - OK, as all terms reduce to (radians). ... actually - issue of dimensionless versus radians??? /********************* /%>>>>>>>>> ∂[∂(t): E0ods(POIo,t)], using proper E0odv(POIo,t) vector approach /*see Figure "Electrostatic field basics & calculus for a POIo", starting with knowledge that the electrostatic field at a point is the same for RFp as for RFo, therefore : /% ∂[∂(t): E0ods(POIo,t)] = ∂[∂(t): E0pds(POIo(t),t)] and : (mathH) ∂[∂(t): E0pds(POIo(t),t)] = ∂[∂(t): |E0pdv(POIo(t),t)|] (endMath) /*+-----+ (RFp) FORMAT /%From "E0odv(POIo,t) = E0pdv(POIo(t),t)" : 1*) E0pdv(POIo(t),t) = E0odv(POIo,t) = Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) Subbing (1)* into (1) : (1) ∂[∂(t): E0pds(POIo(t),t)] = ∂[∂(t): |E0pdv(POIo(t),t)|] = ∂[∂(t): |Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t)|] (2) = |Q(PART)| *dot* ∂[∂(t): |Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2|] /*From "Howell's use of Kahan vector derivative formulation" : (1)** ∂[∂(t): |z|] = z_T *dot* ∂[∂(t):z] / |z| where *dot* = dotProduct where : z, ∂[∂(t):z] are column vectors, z_T is a row vector _T indicates [vector, matrix] transpose * is matrix multiplication (here z_T*∂[∂(t):z] is same as dotProduct as yields 1row*1column result |...| denotes norm question: can one assume that "z_T" is the linear functional dual to z wrt |z|? Applying the template (1)** to the derivative of (2) : /%(3) ∂[∂(t): |Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2|] = [ Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2 ]_T *dot* ∂[∂(t): Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2] / | Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2 | /*+--+ Now looking at : /%(4) ∂[∂(t): Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2] = ∂[∂(t): Rpch(POIo(t),t)]/Rpcs(POIo(t),t)^2 + Rpch(POIo(t),t)*∂[∂(t): Rpcs(POIo(t),t)^( - 2)] First derivative term from "∂[∂(t): Rpch(POIo(t),t)]" Equation (2) : (2)* ∂[∂(t): Rpch(POIp)] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*RDEpdh(POIo(t),t) where RDEpdh(POIo(t),t) is anchored at end of Rpch(POIo(t),t) and is at angle Aθpc(POIo(t),t) + PI/2, ie perpendicular to Rpch(POIo(t),t) angle Aφpc(POIo(t),t) doesn't change Second term's derivative : (5) ∂[∂(t): Rpcs(POIo(t),t)^( - 2)] = -2*Rpcs(POIo(t),t)^(-3)*∂[∂(t): Rpcs(POIo(t),t)] Subbing (2)* & (5) into (4) : (4) ∂[∂(t): Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2] = ∂[∂(t): Rpch(POIo(t),t)]/Rpcs(POIo(t),t)^2 + Rpch(POIo(t),t)*∂[∂(t): Rpcs(POIo(t),t)^( - 2)] = ∂[∂(t): Rpch(POIo(t),t)] /Rpcs(POIo(t),t)^2 + Rpch(POIo(t),t) *∂[∂(t): Rpcs(POIo(t),t)^( - 2)] (6) = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*RDEpdh(POIo(t),t) /Rpcs(POIo(t),t)^2 + Rpch(POIo(t),t) *-2*Rpcs(POIo(t),t)^(-3) *∂[∂(t): Rpcs(POIo(t),t)] From "∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|]" : (1)* ∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|] = -Vons(PART)*cos(Aθpc(POIo(t),t)) Subbing (1)* into (6) : ∂[∂(t): Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*RDEpdh(POIo(t),t) /Rpcs(POIo(t),t)^2 + Rpch(POIo(t),t) *-2*Rpcs(POIo(t),t)^(-3) *-Vons(PART)*cos(Aθpc(POIo(t),t)) (7) = [ Vons(PART)*sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),t) + 2*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) ]/Rpcs(POIo(t),t)^3 /*+--+ Subbing (7) into (3) : (3) ∂[∂(t): |Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2| ] = [ Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2 ]_T *dot* ∂[∂(t): Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2 ] / | Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2 | Keeping the order listed of vectors : = [ Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2 ]_T *dot* [ Vons(PART)*sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),t) + 2*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) ] / Rpcs(POIo(t),t)^3 / | Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2 | = Rpch(POIo(t),t)_T / Rpcs(POIo(t),t)^2 * Vons(PART) *dot* [ sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),t) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) ] / Rpcs(POIo(t),t)^3 / | Rpch(POIo(t),t) | * Rpcs(POIo(t),t)^2 =* [ sin(Aθpc(POIo(t),t))*Rpch(POIo(t),t)_T *dot* RDEpdh(POIo(t),t) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t)_T *dot* Rpch(POIo(t),t) ] * Vons(PART) / | Rpch(POIo(t),t) | / Rpcs(POIo(t),t)^2 / Rpcs(POIo(t),t)^3 * Rpcs(POIo(t),t)^2 (8) =* [ sin(Aθpc(POIo(t),t))*Rpch(POIo(t),t)_T *dot* RDEpdh(POIo(t),t) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t)_T *dot* Rpch(POIo(t),t) ] * Vons(PART) / | Rpch(POIo(t),t) | / Rpcs(POIo(t),t)^3 But : (9) | Rpch(POIo(t),t) | = 1, from "∂[∂(t): Rpch(POIo(t),t)]" (see above) : RDEpdh(POIo(t),t) is anchored at end of Rpch(POIo(t),t) and is at angle Aθpc(POIo(t),t) + PI/2, ie perpendicular to Rpch(POIo(t),t) Rpch(POIo(t),t)_T *dot* RDEpdh(POIo(t),t), where *dot* => dotProduct (see Kreyszig p200) BUT - is v_T perpendicular to v???? ?NO? - no effect here? = |Rpch(POIo(t),t)| * |RDEpdh(POIo(t),t)|*cos(Aθpc(POIo(t),t) + PI/2 - Aθpc(POIo(t),t)) = |Rpch(POIo(t),t)| * |RDEpdh(POIo(t),t)|*cos(PI/2) = |Rpch(POIo(t),t)| * |RDEpdh(POIo(t),t)|*0 (10) = 0 For : Rpch(POIo(t),t)_T *dot* Rpch(POIo(t),t) = |Rpch(POIo(t),t)| * |Rpch(POIo(t),t)|*cos(0) = 1 * 1 * 1 (11) = 1 Subbing (9),(10),(11) into (8) : ∂[∂(t): |Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2| ] =* [ sin(Aθpc(POIo(t),t))*Rpch(POIo(t),t)_T*RDEpdh(POIo(t),t) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t)_T*Rpch(POIo(t),t) ] * Vons(PART) / | Rpch(POIo(t),t) | / Rpcs(POIo(t),t)^3 =* [ sin(Aθpc(POIo(t),t))*0 + 2*cos(Aθpc(POIo(t),t))*1 ] * Vons(PART) / 1 / Rpcs(POIo(t),t)^3 = 2*cos(Aθpc(POIo(t),t))*Vons(PART)/Rpcs(POIo(t),t)^3 /*Key intermediate result useful elsewhere : /%(12) ∂[∂(t): |Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2|] = 2*cos(Aθpc(POIo(t),t))*Vons(PART)/Rpcs(POIo(t),t)^3 /*Subbing (12) into (2) : /% (2) ∂[∂(t): E0pds(POIo(t),t),t] = |Q(PART)|*∂[∂(t): |Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2|] = |Q(PART)|*2*cos(Aθpc(POIo(t),t))*Vons(PART)/Rpcs(POIo(t),t)^3 = 2*|Q(PART)|*cos(Aθpc(POIo(t),t))*Vons(PART)/Rpcs(POIo(t),t)^3 /*Summarizing : /% (mathH) ∂[∂(t): E0ods(POIo,t)] = ∂[∂(t): E0pds(POIo(t),t)] = 2*|Q(PART)|*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 (endMath) Note : This is the SAME as the Rpch component of " ∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)]" : 6*) ∂[∂(t): E0pdv(POIo(t),t)] = Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[ sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) ] where : Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 This should be expected, as per the Figure "∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)]". /*-----+ LIMIT CHECKS /*--+ Does this make sense? /*POIo on line of trajectory of particle as a limit case : Over delta_time of dt, E changes by : initial : Ei = Q / R(t)^2 end : Et = Q / R(t+dt)^2 where R(t+dt) = R(t) + V*dt so dE = Et - Ei = Q*(1/R(t+dt)^2 - 1/R(t)^2) = Q*(1/[R(t) + V*dt]^2 - 1/R(t)^2) = Q*{ R(t)^2 - [R(t) + V*dt]^2 } / { [R(t) + V*dt]^2 * R(t)^2) } = Q*{ R(t)^2 - R(t)^2 - 2*R(t)*V*dt - (V*dt)^2 } / { [R(t)^2 + 2*R(t) *V*dt + (V*dt)^2] * R(t)^2)] = Q*{ - 2*R(t)*V*dt - (V*dt)^2 } / { R(t)^4 + 2*R(t)^3*V*dt + (V*dt)^2 * R(t)^2 } Trick, for very small dt, ignore squared terms " = Q*{ - 2*R(t)*V*dt } / { R(t)^4 + 2*R(t)^3*V*dt } = -2*Q*R(t)*V*dt / R(t)^2 / [ R(t)^2 + 2*R(t)*V*dt ] Also, denominator - ignore dt term = -2*Q*R(t)*V*dt / R(t)^2 / R(t)^2 = -2*Q*R(t)*V*dt / R(t)^4 = -2*Q *V*dt / R(t)^3 So 13a) d[dt : E] = -2*Q*V/R(t)^3 This is the same as (13) except for minus sign, and cosine term Sign - when R < 0, cos < 0, so (13) is +ve, and (13a) is +ve when R > 0, cos > 0, so (13) is -ve, and (13a) is -ve OK!! /*--+ Second check, take tp when POIp is perpendicular to particle : dE = Et - Ei = Q*(1/R(t+dt)^2 - 1/R(t)^2), for absolute values of Q, R = Q*(1/{ [R(t)^2 + (V*dt)^2]^(1/2) }^2 - 1/R(t)^2) = Q*(1/{ [R(t)^2 + (V*dt)^2] } - 1/R(t)^2) = Q*(1/{ [R(t)^2 + (V*dt)^2] } - 1/R(t)^2) /*--+ Dimensional check /% (13) ∂[∂(t): E0ods(POIo,t)] = ∂[∂(t): E0pds(POIo(t),t)] = 2*|Q(PART)|*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 = charge * length/time / length^3 = charge / time / length^2 /*From "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt" : dimensions of E = charge/length^2 therefore d[dt : E] units are charge/length^2/time This is OK /*+-----+ (RFo) FORMAT From "Rpcs(POIo(t),t)" : /% (3)* Rpcs(POIo(t),t) = |Rpcv(POIo(t),t)| = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) From "cos(Aθpc(POIo(t),t))" : (1)* cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) Subbing (3)*&(1)* into (13) : (13) ∂[∂(t): E0pds(POIo(t),t)] = 2*|Q(PART)|*cos(Aθpc(POIo(t),t))*Vons(PART)/Rpcs(POIo(t),t)^3 = 2*|Q(PART)|*Vons(PART) *cos(Aθpc(POIo(t),t)) /Rpcs(POIo(t),t)^3 = 2*|Q(PART)|*Vons(PART) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^3 = 2*|Q(PART)|*Vons(PART) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^2 Summarizing : (mathH) ∂[∂(t): E0ods(POIo,t)] = ∂[∂(t): E0pds(POIo(t),t)] = 2*|Q(PART)|*Vons(PART) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^2 (endMath) /*+--+ LIMIT CHECKS : Dimensional consistency for magnitude - OK, as all units reduce to (charge/length^2/t) Dimensional consistency for direction - OK, as all terms reduce to (radians). ... actually - issue of dimensionless versus radians??? /********************* >>>>>>>>> ∂[∂(t): E0ods(POIo,t)] = ∂[∂(t): E0pds(POIo(t),t)] - cheating E0ods(POIo,t) scalar approach WARNING : Should do vector differentiation THEN take the magnitude!?? -> this was done above (after the cheating approach) Given the parametric equation for E0ds(POIo(t),t), perhaps this isn't a problem? Actually, it WORKS! see Figure "∂[∂(t): E0odv(POIo,t)] - basic metrics" /%From "E0ods(POIo,t) = E0pds(POIo(t),t)" : (2) E0ods(POIo,t) = |Q(PART)|/Rpcs(POIp(t),t)^2 and Rpch is a unit vector, so : (1) |Rpch(POIp(t),t)| = 1 So : (2) ∂[∂(t): E0ods(POIo,t)] = ∂[∂(t): |Q(PART)|/Rpcs(POIp(t),t)^2] = |Q(PART)|/Rpcs(POIp(t),t)*∂[∂(t): Rpcs(POIp(t),t)^( - 3)] From "∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)]" : (1) ∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|] = -Vons(PART)*cos(Aθpc(POIo(t),t)) Subbing (1)* into (2) : (3) ∂[∂(t): E0pds(POIo(t),t)] = ∂[∂(t): |Q(PART)|/Rpcs(POIp(t),t)^2] = |Q(PART)|*(-2)/Rpcs(POIp(t),t)^3*-Vons(PART)*cos(Aθpc(POIo(t),t)) /*For now - ARBITRARY sign on equations for scalar derivatives, as I have to go back & fix the derivations (not as straightforward as it sounds!). /%∂[∂(t): EIods(POIo(t),t)] has the opposite sign of ∂[∂(t): E0ods(POIo,t)] Note that for : /% -PI/2 <= Aθpc(POIo(t),t)) <= PI/2 : then cos(Aθpc(POIo(t),t)) >0 and ∂[∂(t): E0pds(POIo(t),t)] < 0, and ∂[∂(t): EIpds(POIo(t),t)] < 0 /*19Dec2017 The following is nonsensical, as PI/2 >= -PI/2 !!?? /% PI/2 <= Aθpc(POIo(t),t)) <= -PI/2 : then cos(Aθpc(POIo(t),t)) <0 and ∂[∂(t): E0pds(POIo(t),t)] > 0, and ∂[∂(t): EIpds(POIo(t),t)] > 0 /*For Vonv(PART) > 0 (as per definition of BOTH coordinate frames of reference!), these results make sense Finally : /% (mathH) ∂[∂(t): E0pds(POIo(t),t)] = 2*|Q(PART)|*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIp(t),t)^3 (endMath) /*Actually, this cheating approach WORKS! /*+-----+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (charge/length^2/time). (ignoring electric permeability for Gaussian coordinates) Check scalar versus vector approaches : /%From "∂[∂(t): E0ods(POIo,t)] = ∂[∂(t): E0pds(POIo(t),t)] - proper E0odv(POIo,t) vector approach" : (3) ∂[∂(t): E0pds(POIo(t),t)] = |Q(PART)|*Vons(PART)/Rpcs(POIo(t),t)^3*[1 + 3*cos(Aθpc(POIo(t),t))^2] /*This is interesting as the difference between the vector and scalar(incorrect) approaches is in the constant coefficients and cos terms : /%Vector : [1 + 3*cos(Aθpc(POIo(t),t))^2] which is in the interval [1,4] /*Scalar : 2 /*+-----+ Reminders for Chapter 4 : OBJECTIVES : Where convenient, express variables as functions of FIXED variables : (POIp) -> -Vodv(PART),t,Rpcv(POIp),Aθpc(POIp),Aφpc(POIp) (POIo) -> Vodv(PART),t,Rocv(POIo),Aθoc(POIo),Aφoc(POIo) /*From "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)" : /% (2) Rpcs(POIo(t),t)) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /*From "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)" : /% (6) cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (2)* and (6)* into (4) : (4) ∂[∂(t): E0ods(POIo,t)] = 2*|Q(PART)|*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIp)^3 = 2*|Q(PART)|*Vons(PART) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^3 = 2*|Q(PART)|*Vons(PART) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 /*For now - ARBITRARY sign on equations for scalar derivatives, as I have to go back & fix the deerivations (not as straightforward as it sounds!). ∂[∂(t): EIods(POIo(t),t)] has the opposite sign of ∂[∂(t): E0ods(POIo,t)] Note that for : /% -PI/2 <= Aθpc(POIo(t),t)) <= PI/2 : then cos(Aθpc(POIo(t),t)) >0 and ∂[∂(t): E0pds(POIo(t),t)] < 0, and ∂[∂(t): EIpds(POIo(t),t)] < 0 /*19Dec2017 The following is nonsensical, as PI/2 >= -PI/2 !!?? /% PI/2 <= Aθpc(POIo(t),t)) <= -PI/2 : then cos(Aθpc(POIo(t),t)) <0 and ∂[∂(t): E0pds(POIo(t),t)] > 0, and ∂[∂(t): EIpds(POIo(t),t)] > 0 /*For Vonv(PART) > 0 (as per definition of BOTH coordinate frames of reference!), these results make sense Summarizing : /% (mathH) ∂[∂(t): E0ods(POIo,t)] = 2*|Q(PART)|*Vons(PART) *{ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 (endMath) /*Actually, it WORKS! /*+--+ LIMIT CHECKS : see "proper vector approach" above these derivations /********************* >>>>>>>>> Figure "∂[∂(t): BTodv(POIo,t)]" http://www.BillHowell.ca/ /media/bill/SWAPPER/Lucas - Universal Force/Images/Howell - Chapter 4 - dp[dt _ BTodv(POIo,t)] - cropped.png /********************* /%>>>>>>>>> ∂[∂(t): BTodv(POIp(t),t)] = ∂[∂(t): BTpdv(POIo(t),t)] ≠ ∂[∂(t): BTpdv(POIp)] = 0, without use of Lenz's Induction Law (need to RE-CHECK!!!) /*<<< 29Mar2018 Is it correct that ∂[∂(t): BTpdv(POIo(t),t)] ≠ 0 ? >>> <<< 30Mar2018 Must I do spherical coordinate derivatives, as did Lucas? >>> In order to compare to Lucas's intermediate results, it is handy to have an expression BEFORE invoking of Lenz's Law. Looking at Figure "∂[∂(t): BTodv(POIo,t)] = ∂[∂(t): BTpdv(POIo(t),t)]" : From "Differential geometry of ∂[∂(t): BTodv(POIo,t)] = ∂[∂(t): BTpdv(POIo(t),t)]" : /% (mathH) ∂[∂(t): E0pdv(POIo(t),t)] = E0pds(POIo(t),t)*∂[∂(t): Aθpc(POIo(t),t)]*RDEpdh(POIo(t),∂(t)) + ∂[∂(t): |E0pdv(POIo(t),t)|] *Rpch(POIo(t),t) (endMath) /*... ignoring differential change in direction for RDEpdh(POIo(t),dt) & Rpch(POIo(t),t) /*+-----+ (RFp) basis <<< 30Mar2018 Right now this does simple planar derivative (Cartesian), not spherical coordinates as Lucas did >>> /%From "BTodv(POIo,t) = BTpdv(POIo(t),t), without use of Lenz's Law" : (6) BTpdv(POIo(t),t) = BTodv(POIo,t) = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) * [ Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t) ] where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Therefore, taking the derivative of (6)* : ∂[∂(t): BTpdv(POIo(t),t)] = ∂[∂(t): Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t))*[Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t)]] = ∂[∂(t): Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t))]* [ Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t)] + Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *∂[∂(t): Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t)] (1) = ∂[∂(t): Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t))]* [ Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t)] + Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t))*{∂[∂(t): Q(PART)/Rpcs(POIo(t),t)^2]- ∂[∂(t): EIpds(POIo(t),t)] } /*+--+ /%Looking at the term in (1) ∂[∂(t): Rodh(Vonv_X_Rpcv(POIo(t),t))] From Figure "∂[∂(t): Rodh(Vonv_X_Rpcv(POIo(t),t))]" /*As with most of the models for Chapter 4, the phi (P) line-of-direction of the magnetic field B and electric field E of a [particle, POIo, POIp] does not change with time. Additionally, while the the theta (O) direction does change with time for E for a POIo (but NOT, or course, for the [particle, POIp] as [E,B] = constant = 0 for those situations), it does NOT change for B, as the latter is perpendicular to the [Vonv, Rpcv] plane, which itself is of constant P (phi) direction. Therefore, using /% Aφoc(Vonv&Rpcv(POIo(t),t)) = Angle between the plane of Vonv_X_Rpcv(POIo(t),t), and Aφod(BTodv(POIo,t)): /% Aφoc(planeofVonv&Rpcv(POIo(t),t)) = constant = Aφoc(POIo) AngleBetween(planeofVonv&Rpcv(POIo(t),t),APod(BTodv(POIo,t))) = constant = PI/2 in phi (i.e. Aφpd(Vonv_X_Rpcv(POIo(t),t))) Aφoc(BT(POIo(t),t)) = constant = Aφoc(POIo) + PI/2 (again assuming BI,BT are collinear) Neither the line-of-direction nor magnitude (obviously as this is a unit vector) of Rodh(Vonv_X_Rpcv(POIo(t),t)) change. However, while the LINE-OF-DIRECTION of Rodh(Vonv_X_Rpcv(POIo(t),t)) does not change, its sign CAN change depending on sin(Vonv_X_Rpcv(POIo(t),t)), which is just sin(Aθpc(POIo(t),t))!! Therefore : (2) ∂[∂(t): Rodh(Vonv_X_Rpcv(POIo(t),t))] = sign[sin(Vonv_X_Rpcv(POIo(t),t))] = [ - 1,1] Denote : (3) sgn(BV) = ∂[∂(t): Rodh(Vonv_X_Rpcv(POIo(t),t))] = sign[sin(Vonv_X_Rpcv(POIo(t),t))] = [ - 1,1] /*WRONG!!! - should do properly... Therefore this is a "kind-of-constant" and it can be moved outside from the term /%∂[∂(t): sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t))]. /*+--+ /%Bringing constants [ Q(PART),Vons(PART)/c, Rodh(Vonv_X_Rpcv(POIo(t),t))] out of the derivatives of (1), but use sgn(BV)*Rodh(Vonv_X_Rpcv(POIo(t),t)) when *Rodh(Vonv_X_Rpcv(POIo(t),t)) taken out sign(: (1) ∂[∂(t): BTpdv(POIo(t),t)] = ∂[∂(t): Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t))]*[ Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t)] + Vons(PART)/c*sin(Aθpc(POIo(t),t)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ ∂[∂(t): Q(PART)/Rpcs(POIo(t),t)^2] - ∂[∂(t): EIpds(POIo(t),t)] } = Vons(PART)/c*∂[∂(t): sin(Aθpc(POIo(t),t))]*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t)] + Vons(PART)/c *sin(Aθpc(POIo(t),t)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) *{Q(PART)*∂[∂(t): 1/Rpcs(POIo(t),t)^2] - ∂[∂(t): EIpds(POIo(t),t)] } (3) = Vons(PART)/c *Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ ∂[∂(t): sin(Aθpc(POIo(t),t))] *[Q(PART) /Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t) ] + sin(Aθpc(POIo(t),t)) *[Q(PART)*∂[∂(t): 1/Rpcs(POIo(t),t)^2]- ∂[∂(t): EIpds(POIo(t),t)] ] } /*+--+ Looking at : (4) ∂[∂(t): 1/Rpcs(POIo(t),t)^2 ] = ∂[∂(t): Rpcs(POIo(t),t)^(-2) ] = -2*Rpcs(POIo(t),t)^(-3)*∂[∂(t): Rpcs(POIo(t),t)] From "∂[∂(t): Rpcs(POIo(t),t) ] = ∂[∂(t): |Rpcv(POIo(t),t)|]" : (1) ∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)| ] = -Vons(PART)*cos(Aθpc(POIo(t),t)) Subbing (1)* into (4) : (4) ∂[∂(t): 1/Rpcs(POIo(t),t)^2 ] = ∂[∂(t): Rpcs(POIo(t),t)^(-2) ] = -2*Rpcs(POIo(t),t)^(-3)*∂[∂(t): Rpcs(POIo(t),t)] = -2*Rpcs(POIo(t),t)^(-3)*-Vons(PART)*cos(Aθpc(POIo(t),t)) (5) = 2*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 From "∂[∂(t): sin(Aθpc(POIo(t),t)) ]" : (2)* ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) /*+--+ Subbing (5) & (2)* into (3) : (3) ∂[∂(t): BTpdv(POIo(t),t)] = ∂[∂(t): BTodv(POIo,t)] = Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo)) *{ ∂[∂(t): sin(Aθpc(POIo(t),t))]*[ Q(PART) /Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t) ] + sin(Aθpc(POIo(t),t)) *[ Q(PART) *∂[∂(t): 1/Rpcs(POIo(t),t)^2]- ∂[∂(t): EIpds(POIo(t),t)] ] } = Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo)) *{ Vons(PART) *sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) *[ Q(PART) /Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t)] + sin(Aθpc(POIo(t),t))*[ Q(PART)*2*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 - ∂[∂(t): EIpds(POIo(t),t)] ] } = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo)) *{ Q(PART)*Vons(PART) *sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 - Vons(PART) *sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) *EIpds(POIo(t),t) + 2*Q(PART)*Vons(PART) *sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 - sin(Aθpc(POIo(t),t)) *∂[∂(t): EIpds(POIo(t),t)] ] } = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo)) *Vons(PART) *sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t)) *{ Q(PART)/Rpcs(POIo(t),t)^3 - /Rpcs(POIo(t),t) *EIpds(POIo(t),t) + 2*Q(PART)/Rpcs(POIo(t),t)^3 } - Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo))*sin(Aθpc(POIo(t),t))*∂[∂(t): EIpds(POIo(t),t)] = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo)) *{ 3*Q(PART)/Rpcs(POIo(t),t)^3 - EIpds(POIo(t),t)/Rpcs(POIo(t),t) } - Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo))*sin(Aθpc(POIo(t),t))*∂[∂(t): EIpds(POIo(t),t)] ={ 3*Q(PART)/Rpcs(POIo(t),t)^3*Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t)) - Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t)) *EIpds(POIo(t),t)/Rpcs(POIo(t),t) - Vons(PART) /c*sin(Aθpc(POIo(t),t))^2 *∂[∂(t): EIpds(POIo(t),t)] } *Rodh(Vonv_X_Rpcv(POIo)) ={ 3*Q(PART)*Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 - Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) *EIpds(POIo(t),t) - Vons(PART) /c*sin(Aθpc(POIo(t),t))^2 *∂[∂(t): EIpds(POIo(t),t)] } *Rodh(Vonv_X_Rpcv(POIo)) = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo)) *{ 3*Q(PART) /Rpcs(POIo(t),t)^3 - EIpds(POIo(t),t)/Rpcs(POIo(t),t) - ∂[∂(t): EIpds(POIo(t),t)]/Vons(PART)/cos(Aθpc(POIo(t),t)) } Summarising : (mathH) ∂[∂(t): BTpdv(POIo(t),t)] = ∂[∂(t): BTodv(POIo,t)] = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo)) *{3*Q(PART)/Rpcs(POIo(t),t)^3 - EIpds(POIo(t),t)/Rpcs(POIo(t),t) - ∂[∂(t): EIpds(POIo(t),t)]/Vons(PART)/cos(Aθpc(POIo(t),t))} (endMath) <<<< Improper derivative of Rodh(Vonv_X_Rpcv(POIo)) ]but use sgn(BV)*Rodh(Vonv_X_Rpcv(POIo)) when *Rodh(Vonv_X_Rpcv(POIo)) taken out /*At this stage (pre [Lenz's Law & Thomas Barnes iterations]), terms with EIpds(POIo(t),t) stay the same. /*+--+ Limit checks : 29Mar2018 Dimensional check Subbing in units of measure : /%(6)* ∂[∂(t): BTpdv(POIo(t),t)] = ∂[∂(t): BTodv(POIo,t)] (charge/length/time)/time = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) (length/time )^2 *{ 3*Q(PART) /Rpcs(POIo(t),t)^3 charge / length^3 - EIpds(POIo(t),t)/Rpcs(POIo(t),t) (charge/length^2) / length - ∂[∂(t): EIpds(POIo(t),t)]/Vons(PART)/cos(Aθpc(POIo(t),t)) (charge/length^2/time) /(length/time) } = (length/time )^2 *{ charge/length^3 - (charge/length^2) / length - (charge/length^2/time) /(length/time) } = (length/time )^2 *{ charge/length^3 - (charge/length^3) - (charge/length^3) } = (length/time )^2 * charge/length^3 = (charge/length/time^2) /*Therefore (charge/length/time^2) = (charge/length/time^2) = ∂[∂(t): B] units So the dimensions are consistent. /*+-----+ (RFo) basis /%From "BTodv(POIo,t) = BTpdv(POIo(t),t) , without use of Lenz's Law" : (6) BTpdv(POIo(t),t) = BTodv(POIo,t)] = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART)/Rpcs(POIo(t),t)^2 - EIods(POIo(t),t) ] where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) in RFp coordinates /*OOOPS!!! - I should have taken : /% (9) BTpdv(POIo(t),t) = BTodv(POIo,t) = Vons(PART)/c *Rocs(POIo)*sin(Aθoc(POIo)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) - EIods(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ] where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Anyways, still proceeding from (6)* : ∂[∂(t): BTodv(POIo,t)] = ∂[∂(t): Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t))*[Q(PART)/Rpcs(POIo(t),t)^2 - EIods(POIo(t),t)]] = ∂[∂(t): Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t))] *[ Q(PART)/Rpcs(POIo(t),t)^2 - EIods(POIo(t),t) ] + Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *∂[∂(t): Q(PART)/Rpcs(POIo(t),t)^2 - EIods(POIo(t),t)] = ∂[∂(t): Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t))] *[ Q(PART)/Rpcs(POIo(t),t)^2 - EIods(POIo(t),t) ] + Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ ∂[∂(t): Q(PART)/Rpcs(POIo(t),t)^2] - ∂[∂(t): EIods(POIo(t),t)] ] (1) = Vons(PART)/c* { ∂[∂(t): sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t))] *[ Q(PART) /Rpcs(POIo(t),t)^2 - EIods(POIo(t),t) ] + sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART)*∂[∂(t): 1/Rpcs(POIo(t),t)^2]- ∂[∂(t): EIods(POIo(t),t)] ] } Removing Rodh(Vonv_X_Rpcv(POIo(t),t)) from within derivatives as a constant, Equation (1) becomes : (1) BTodv(POIo,t)] = Vons(PART)/c* { ∂[∂(t): sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t))] *[ Q(PART) /Rpcs(POIo(t),t)^2 - EIods(POIo(t),t) ] + sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART)*∂[∂(t): 1/Rpcs(POIo(t),t)^2] - ∂[∂(t): EIods(POIo(t),t)] ] } (3) = Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t))* { ∂[∂(t): sin(Aθpc(POIo(t),t))] *[ Q(PART) /Rpcs(POIo(t),t)^2 - EIods(POIo(t),t) ] + sin(Aθpc(POIo(t),t)) *[ Q(PART)*∂[∂(t): 1/Rpcs(POIo(t),t)^2] - ∂[∂(t): EIods(POIo(t),t)] ] } /*+--+ From "∂[∂(t): sin(Aθpc(POIo(t),t)) ]" : (1) ∂[∂(t): sin(Aθpc(POIo(t),t))] = -Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) * [ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] / [ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART) *t)^2 ]^(3/2) Looking at : (4) ∂[∂(t): 1/Rpcs(POIo(t),t)^2 ] = ∂[∂(t): Rpcs(POIo(t),t)^(-2) ] = -2*Rpcs(POIo(t),t)^(-3)*∂[∂(t): Rpcs(POIo(t),t)] From "∂[∂(t): Rpcs(POIo(t),t) ] = ∂[∂(t): |Rpcv(POIo(t),t)|]" : (7) ∂[∂(t): Rpcs(POIo(t),t)] = Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(1/2) /*From "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)" : /% (2) Rpcs(POIo(t),t)) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (7)* & (2)* into (4) : (4) ∂[∂(t): 1/Rpcs(POIo(t),t)^2 ] = ∂[∂(t): Rpcs(POIo(t),t)^(-2) ] = -2*Rpcs(POIo(t),t)^(-3)*∂[∂(t): Rpcs(POIo(t),t)] = -2*{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^(-3) *Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(1/2) = -2*Vons(PART) *{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-3/2) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(1/2) (5) = -2*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 /*+--+ Subbing (1)*,(5)*,(5) into (3) (3) ∂[∂(t): BTodv(POIo,t)] = Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo))* {∂[∂(t): sin(Aθpc(POIo(t),t)) ] *[ Q(PART)/Rpcs(POIo(t),t)^2 - EIods(POIo(t),t) ] + sin(Aθpc(POIo(t),t)) *[ Q(PART)*∂[∂(t): 1/Rpcs(POIo(t),t)^2 ] - ∂[∂(t): EIods(POIo(t),t)] ] } = Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo))* { - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(3/2) *[ Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^2 - EIods(POIo(t),t) ] + sin(Aθpc(POIo(t),t)) *[ Q(PART)*-2*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 - ∂[∂(t): EIods(POIo(t),t)] ] } Summarizing : (mathH)/* Note that I keep the direction vector Rodh(Vonv_X_Rpcv(POIo)) for simplicity!! /% ∂[∂(t): BTodv(POIo,t)] = Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo))* { - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(3/2) *[ Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^2 - EIods(POIo(t),t) ] + sin(Aθpc(POIo(t),t)) *[ Q(PART)*-2*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 - ∂[∂(t): EIods(POIo(t),t)] ] } (endMath) /*+--+ Limit checks : +-+ Dimensional consistency, as a convention here take [unit vectors, c] as dimensionless : (see file "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt" sub-sub-section "Dimensional analysis (Gaussian units)" for dimensional analysis notes) /% (6) ∂[∂(t): BTodv(POIo,t)] = Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t))* { - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(3/2) *[ Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } + EIods(POIo(t),t) ] + sin(Aθpc(POIo(t),t)) *[ Q(PART)*-2*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 + ∂[∂(t): EIods(POIo(t),t)] ] } /* = (length/time) { - (length)*(length/time) *[ - (length) + (length/time)*(time) ] /[ (length)^2 - (length)*(length/time)*(time) + (length/time*time)^2 ]^(3/2) *[ (charge) /{ (length)^2 - (length)*(length/time)*(time) + [(length/time)*(time)]^2 } + (charge/length^2) ] + 1 *[ (charge)*-(length/time) *[ - (length) + (length/time)*(time) ] /{ (length)^2 - (length)*(length/time)*(time) + [(length/time)*(time)]^2 }^2 + (charge/length^2)/(time) ] } = (length/time) { - (length)^2/(time) *[ - (length) + (length) ] /[ (length)^2 - (length)^2 + (length)^2 ]^(3/2) *[ (charge) /{ (length)^2 - (length)^2 + (length)^2 } + (charge/length^2) ] + 1 *[ (charge)*-(length/time) *[ - (length) + (length) ] /{ (length)^2 - (length)^2 + (length)^2 }^2 + (charge/length^2/time) ] } = (length/time) { (length)^2/(time)*(length)/[(length)^2]^(3/2) *[ (charge)/(length)^2 + (charge/length^2) ] + [ (charge)*(length/time)*(length) /(length)^2^2 + (charge/length^2/time) ] } = (length/time) { (length)^(2+1-3)/(time) * (charge)/(length)^2 + [ (charge)*(length)^(1+1-4)/(time) + (charge/length^2/time) ] } = (length/time) { charge/length^2/time - [ charge/length^2/time + charge/length^2/time ] } = charge/length/time^2 From "Dimensional analysis (Gaussian units) ", sub-section "Dimensional analysis (Gaussian units)" : units of ∂[∂(t): BTpdv(POIo(t),t)] = (charge/length/time^2) units OK as all units reduce to (charge/length/time^2) (ignoring permittivity & permeability for Gaussian units) +-+ Reconciliation of RFo and RFp results later .... /********************************************** >>>>>> Lenz's Induction Law - Basics and Calculus 23Mar2016 - Equations were all renumbered, augmented by new equations. Need to change references to these equations in the rest of this document. <<< 24Mar2018 – My vector expressions are WRONG in that vector components must be reversed, not just negated!!! >>> However, the same expression results for simple vector additions, but does this work for [inner, cross] product and other functions? I doubt it... or at least I would have to prove it. IE - In vector notation, does the negative sign imply opposite directions? /********************* >>>>>>>>> Lenz's Induction Law and it's context From Lucas's book, p64h0.4 Equation (4-5) : /%(4-5) EI_LENZodv(POIo,t) ∝ E0ods(POIo,t) * (- Rpch(POIo(t),t) ) → 24Mar2018 changed from E0odv(POIo,t) where ET_LENZodv(POIo) = E0odv(POIo) + EI_LENZodv(POIo) /*From Lucas's book, p70h0.85 Equation (4-31) : → <<< 24Mar2018 YESSS!!! Lucas had that right >>> /% (mathH)/* (4-31) /% EI_LENZods(POIo,t)|(Aθpc(POIo(t),t=0))*Rpch(POIo(t),t) = -λ(Vons(PART))*E0ods(POIo)*Rpch(POIo(t),t) (endMath) where λ(Vons(PART)) is a positive real constant, given that Vonv(PART) is a constant /*<<< It's not clear to me WHY this has been restricted to the condition |(Aθpc(POIo))=0), but in the developments below I will ignore that restiction. >>> /*+-----+ 15May2016 From : 30Mar2016 ?Inconsistency? : Lenz's Induction Law versus Barnes iterations - I have been assuming : ET = E0 + EI = E0 - lambda(v)*E0 = (1 - lambda(v))*E0 = (1 - beta^2)*E0 but in the end Lucas uses : ET = E0*(1-beta^2)/(1 - beta^2*sin(theta)^2)^(3/2) This is not consistent!!! see (4-31) to (4-43) The denominator of that last expression = 1 when (Aθpc(POIo))=0), so perhaps this explains the condition used by Lucas. /*+-----+ /%(1) EI_LENZodv(POIo(t),t) = -λ(Vons(PART))*E0odv(POIo,t) where λ(Vons(PART)) is a positive real function of speed /*Note that in Chapter 4 of Lucas's book, [Q(PART), Vonv(PART), lambda(v), ???] are all constant. For a Point Of Interest (POIo) in the observer reference frame (RFo), ???? AT CONSTANT RELATIVE VELOCITY (POIo, particle) by Lenz's Law (see (4-31)*, then [EIodv(POIo,t),∂[∂(t): EIodv(POIo,t)]] are just scalar multiples of [E0odv(POIo,t),∂[∂(t): E0odv(POIo,t)]]. 12May2016 Remember, this ONLY applies to the constant particle velocity situation as in Chapter 4!!! 12May2016 NOT generally correct!! -> sin(theta) must be taken into account at the surface of a particle, even though here I don't need to as only point-sized particles are being addressed at this stage, so vector EIodv(POIo,t) is just a scalar multiple of EIodv(POIo,t). However, this issue does become a problem for finite-sized particles. 12May2016 p68h0.5 Figure (4-4) Halelujeh!! -- Sometimes, Lucas's angle theta is the angle at the surface of a charged particle between the suface normal vector and B !!! /********************* /%>>>>>>>>> EI_LENZodv(POIo,t) = EI_LENZpdv(POIo(t),t), using Lenz's Induction Law /*From "Lenz's Induction Law and it's context" : /% (1)* EI_LENZodv(POIo(t),t) = -λ(Vons(PART))*E0odv(POIo,t) where λ(Vons(PART)) is a positive real function of speed /*+-----+ (RFp) basis /%From "E0odv(POIo,t) = E0pdv(POIo(t),t)" : (1)** E0pdv(POIo(t),t) = E0odv(POIo,t) = Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) /*Subbing (1)** into (1)* : /% (mathH) EI_LENZpdv(POIo(t),t) = -λ(Vons(PART))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) (endMath) /* where lambda(v) is a positive real function of speed /*+-----+ (RFo) basis From "Rpcs(POIo(t),t)" : /% (2) Rpcs(POIo(t),t) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (2)* into (2) : EI_LENZodv(POIo(t),t) = EI_LENZpdv(POIo(t),t) = -λ(Vons(PART))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) = -λ(Vons(PART))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^2 Finally : (mathH) EI_LENZodv(POIo(t),t) = -λ(Vons(PART))*Q(PART)*Rpch(POIo(t),t) /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2} (endMath) /********************* >>>>>>>>> ET_LENZodv(POIo(t),t) = ET_LENZpdv(POIo(t),t), using Lenz's Induction Law /*+-----+ (RFp) basis Following Lucas p67h0.6 Eqn (4-13) & (4-41) : /% (1)* ET_LENZpdv(POIo(t),t) = E0pdv(POIo(t),t) + EI_LENZpdv(POIo(t),t) From "E0odv(POIo,t) = E0pdv(POIo(t),t)" : (1)** E0pdv(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) From "EI_LENZodv(POIo(t),t) = EI_LENZpdv(POIo(t),t), using Lenz's Induction Law" : (2)* EI_LENZpdv(POIo(t),t) = -λ(Vons(PART))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) Subbing (1)** & (2)* into (1)* : (1)* ET_LENZpdv(POIo(t),t) = E0pdv(POIo(t),t) + EI_LENZpdv(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) + -λ(Vons(PART))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) = (1 - λ(Vons(PART)))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) Summarizing : (mathH) ET_LENZpdv(POIo(t),t) = (1 - λ(Vons(PART)))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) (endMath) /*+-----+ (RFo) basis Following Lucas p67h0.6 Eqn (4-13) & (4-41) : /% (1)* ET_LENZodv(POIo(t),t) = E0odv(POIo,t) + EI_LENZodv(POIo(t),t) From "E0odv(POIo,t) = E0pdv(POIo(t),t)" : (5)* E0odv(POIo) = Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } where Rpch(POIo(t),t) = Rodh(POIo) = displacement vector [start : (POIo), length : 1, theta : arccos(Aθpc(POIo(t),t)), phi : Aφoc(POIo)] and cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) From "EI_LENZodv(POIo(t),t) = EI_LENZpdv(POIo(t),t), using Lenz's Induction Law" : (3)* EI_LENZodv(POIo(t),t) = -λ(Vons(PART))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } Subbing (5)* & (3)* into (1)* : (1)* ET_LENZodv(POIo(t),t) = E0odv(POIo,t) + EI_LENZodv(POIo(t),t) = Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } + -λ(Vons(PART))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } = (1 - λ(Vons(PART)))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } So : (mathH) ET_LENZodv(POIo(t),t) = (1 - λ(Vons(PART)))*Q(PART)*Rpch(POIo(t),t) /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2} (endMath) /********************* >>>>>>>>> EI_LENZods(POIo(t),t) = EI_LENZpds(POIo(t),t), using Lenz's Induction Law (1) EI_LENZods(POIo(t),t) = EI_LENZpds(POIo(t),t) = |EI_LENZodv(POIo(t),t)| = |EI_LENZpdv(POIo(t),t)| For now - ARBITRARY sign on equations for scalar derivatives, as I have to go back & fix the deerivations (not as straightforward as it sounds!). ∂[∂(t): EI_LENZods(POIo(t),t)] has the opposite sign of ∂[∂(t): E0ods(POIo,t)] Note that for : /% -PI/2 <= Aθpc(POIo(t),t)) <= PI/2 : then cos(Aθpc(POIo(t),t)) >0 and ∂[∂(t): E0pds(POIo(t),t)] < 0, and ∂[∂(t): EI_LENZpds(POIo(t),t)] < 0 /*19Dec2017 The following is nonsensical, as PI/2 >= -PI/2 !!?? /% PI/2 <= Aθpc(POIo(t),t)) <= -PI/2 : then cos(Aθpc(POIo(t),t)) <0 and ∂[∂(t): E0pds(POIo(t),t)] > 0, and ∂[∂(t): EI_LENZpds(POIo(t),t)] > 0 /*For Vonv(PART) > 0 (as per definition of BOTH coordinate frames of reference!), these results make sense /*+-----+ (RFp) basis /%From "EI_LENZodv(POIo(t),t) = EI_LENZpdv(POIo(t),t), using Lenz's Induction Law" : 2*) EI_LENZpdv(POIo(t),t) = -λ(Vons(PART))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) /*Subbing (2)* into (1), /% EI_LENZods(POIo(t),t) = |-λ(Vons(PART))*Q/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t)| /* Absolute values of multiplicataion/division series - simply do the absolute value of each term /* Absolute values of addition/subtraction series - cannot just do individual terms - must take whole series I need to fix this!! : From "Scalar absolute values, [vector, matrix] norms - simplification of expressions" : scalar norms - if all terms are scalars - multiplicative/divisive : |product(x1,x2,x3,...)| = product(|x1|,|x2|,|x3|,...) - subtraction/ addition : |sum (x1,x2,x3,...)| ≠ ≠ sum/diff(|x1|,|x2|,|x3|,...) in general vector norms - if some terms are vectors - multiplicative/divisive : |product(x1,x2,x3,...)| = product(|x1|,|x2|,|x3|,...) /% EI_LENZods(POIo(t),t) = |-λ(Vons(PART))*Q/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t)| = |-λ(Vons(PART))|*|Q|/|Rpcs(POIo(t),t)^2|*|Rpch(POIo(t),t)| /*As we know that : - Rpch(POIo(t),t) is a unit vector, so we can replace it by |Rpch(POIo(t),t)| = 1 - lambda(v) is positive (?) - Q(PART) can be positive or negative real, so just tack the norm /% (mathH) EI_LENZpds(POIo(t),t) = λ(Vons(PART))*|Q(PART)|/Rpcs(POIo(t),t)^2 (endMath) /*+-----+ (RFo) basis /%From "Rpcs(POIo(t),t)" : (2) Rpcs(POIo(t),t) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (2)* into (2) : EI_LENZods(POIo(t),t) = EI_LENZpds(POIo(t),t) = λ(Vons(PART))*Q(PART)/Rpcs(POIo(t),t)^2 = λ(Vons(PART))*Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^2 Summarizing : (mathH) EI_LENZods(POIo(t),t) = EI_LENZpds(POIo(t),t) = λ(Vons(PART))*|Q(PART)| /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2} (endMath) /********************* >>>>>>>>> ET_LENZods(POIo(t),t) = ET_LENZpds(POIo(t),t), using Lenz's Induction Law IMPORTANT NOTE : There is an explicit belief in mainstream physics, AND IN LUCAS'S book, an implicit assumption, that speeds can never exceed the speed of light in a vacuum, c. However, that should NOT be a limitation of Lucas's Universal force!! There is therefore no guarantee that v/c is <1, and therefore that : /% Nyet : 0 <= [(1 - λ(Vons(PART))), (1 - β^2) <= 1 /*This affects derivatives! /*+-----+ (RFp) basis /%From "ET_LENZodv(POIo(t),t) = ET_LENZpdv(POIo(t),t), using Lenz's Induction Law" : (1)* ET_LENZpdv(POIo(t),t) = (1 - λ(Vons(PART)))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) By definition : ET_LENZpds(POIo(t),t) = |ET_LENZpdv(POIo(t),t)| = |(1 - λ(Vons(PART)))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t)| = |(1 - λ(Vons(PART)))|*|Q(PART)|/|Rpcs(POIo(t),t)^2|*|Rpch(POIo(t),t)| But [λ(Vons(PART)),Rpcs(POIo(t),t),Rpch(POIo(t),t)] are positive real (& in any case Rpcs(POIo(t),t) is squared). And |Rpch(POIo(t),t)| = 1. So : (mathH) ET_LENZpds(POIo(t),t) = |(1 - λ(Vons(PART)))|*|Q(PART)|/Rpcs(POIo(t),t)^2 (endMath) /*+-----+ (RFo) basis /%From "ET_LENZodv(POIo(t),t) = ET_LENZpdv(POIo(t),t), using Lenz's Induction Law" : (2)* ET_LENZodv(POIo(t),t) = (1 - λ(Vons(PART)))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } /*By definition : /% ET_LENZods(POIo(t),t) = |ET_LENZodv(POIo(t),t)| = | (1 - λ(Vons(PART)))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } | = |(1 - λ(Vons(PART)))|*|Q(PART)|*|Rpch(POIo(t),t)| /|{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }| But [λ(Vons(PART)),Rpcs(POIo(t),t),Rpch(POIo(t),t)] are positive real (& in any case Rpcs(POIo(t),t) is squared). And |Rpch(POIo(t),t)| = 1. So : (mathH) ET_LENZods(POIo(t),t) = |ET_LENZodv(POIo(t),t)| = |(1 - λ(Vons(PART)))|*|Q(PART)| /|{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}| (endMath) /********************* >>>>>>>>> BT_LENZodv(POIo(t),t) = BT_LENZpdv(POIo(t),t), using Lenz's Induction Law Here I will simply adapt results in "BT_LENZodv(POIo(t),t) = BT_LENZpdv(POIo(t),t), without use of Lenz's Law". /*+-----+ (RFp) basis /%From "BT_LENZodv(POIo(t),t) = BT_LENZpdv(POIo(t),t), without use of Lenz's Law" : (6) BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t) = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART)/Rpcs(POIo(t),t)^2 + EI_LENZpds(POIo(t),t) ] where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) From "EI_LENZodv(POIo(t),t) = EI_LENZpdv(POIo(t),t), using Lenz's Induction Law" : (2) EIodv(POIo) = EI_LENZpdv(POIo(t),t) = -λ(Vons(PART))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) Subbing (2)* into (6)* : BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t) = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART)/Rpcs(POIo(t),t)^2 + EI_LENZpds(POIo(t),t) ] = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART)/Rpcs(POIo(t),t)^2 - λ(Vons(PART))*Q(PART)/Rpcs(POIo(t),t)^2 ] = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *Q(PART)/Rpcs(POIo(t),t)^2*(1 - λ(Vons(PART))) Summarizing : (mathH) BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t) = (1-λ(Vons(PART)))*Q(PART)*Vons(PART)/c*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^2*Rodh(Vonv_X_Rpcv(POIo(t),t)) (endMath) where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) /*+-----+ (RFo) basis /%From "Rpcs(POIo(t),t)" : (2) Rpcs(POIo(t),t) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) From "sin(Aθpc(POIo(t),t))" : (5) sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (2)*,(5)* into (1) : (1) BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t) = [1-λ(Vons(PART))]*Q(PART)*Vons(PART)/c*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^2*Rodh(Vonv_X_Rpcv(POIo(t),t)) = [1-λ(Vons(PART))]*Q(PART)*Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) *sin(Aθpc(POIo(t),t)) /Rpcs(POIo(t),t)^2 = [1-λ(Vons(PART))]*Q(PART)*Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) *Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^2 Summarizing : (mathH)/* is there a mistake here? /% BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t) = [1-λ(Vons(PART))]*Q(PART)*Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) *Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) (endMath) /********************* >>>>>>>>> ∂[∂(t): EI_LENZodv(POIo(t),t)] = ∂[∂(t): EI_LENZpdv(POIo(t),t)] - using Lenz's Law, proper vector approach /%From "EI_LENZodv(POIo(t),t) = EI_LENZpdv(POIo(t),t), using Lenz's Induction Law" : (1) EIodv(POIo) = EI_LENZpdv(POIo(t),t) = -λ(Vons(PART))*E0odv(POIo) Therefore : ∂[∂(t): EIodv(POIo)] = ∂[∂(t): EI_LENZpdv(POIo(t),t)] = ∂[∂(t): - λ(Vons(PART))*E0odv(POIo)] Or : (mathH) ∂[∂(t): EIodv(POIo)] = ∂[∂(t): EI_LENZpdv(POIo(t),t)] = -λ(Vons(PART))*∂[∂(t): E0odv(POIo)] (endMath) /*+-----+ (RFp) basis /%From "∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)]" : (6) ∂[∂(t): E0pdv(POIo(t),t)] = Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[ sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) ] where : Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 Multiplying (6)* by -λ(Vons(PART)) : ∂[∂(t): EI_LENZpdv(POIo(t),t) = -λ(Vons(PART))*∂[∂(t): E0pdv(POIo(t),t)] Yielding : (mathH)/* where : λ(Vons(PART)) is a positive constant, Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t), RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 /% ∂[∂(t): EI_LENZpdv(POIo(t),t)] = -λ(Vons(PART))*Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t)] (endMath) /*+-----+ (RFo) basis /%From "∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)]" : (9)* ∂[∂(t): E0pdv(POIo(t),t)] = Q(PART)*Vons(PART) *[ Rocs(POIo)*sin(Aθoc(POIo)) *RDEpdh(POIo(t),∂(t)) + 2*[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rpch(POIo(t),t) ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 where : Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 Aθpc(POIo(t),t) = arcsin [ Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ] = arccos [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) ] Therefore, multiplying (9)* by -λ(Vons(PART)) : (1) ∂[∂(t): EI_LENZodv(POIo(t),t) = -λ(Vons(PART))*∂[∂(t): E0pdv(POIo(t),t)] from "sin(Aθpc(POIo(t),t))" : (5)* sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) from "cos(Aθpc(POIo(t),t))" : (1)* cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) Subbing (1),(5)*,(1)* into (9)* (9)* ∂[∂(t): EI_LENZodv(POIo(t),t)] = -λ(Vons(PART))*Q(PART)*Vons(PART) *[ Rocs(POIo)*sin(Aθoc(POIo)) *RDEpdh(POIo(t),∂(t)) + 2*[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rpch(POIo(t),t) ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 So : (mathH)/* where : λ(Vons(PART)) is a positive constant, Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t), RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 /% ∂[∂(t): EI_LENZodv(POIo(t),t)] = -λ(Vons(PART))*Q(PART)*Vons(PART) *[ Rocs(POIo)*sin(Aθoc(POIo)) *RDEpdh(POIo(t),∂(t)) + 2*[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rpch(POIo(t),t) ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 (endMath) /********************* /%>>>>>>>>> ∂[∂(t): EI_LENZods(POIo(t),t)] = ∂[∂(t): EI_LENZpds(POIo(t),t)] - using Lenz's Law, based on proper vector approach /*+-----+ (RFp) basis /%From "∂[∂(t): EI_LENZodv(POIo(t),t)] = ∂[∂(t): EI_LENZpdv(POIo(t),t)] - using Lenz's Law, proper vector approach" : (2)* ∂[∂(t): EI_LENZpdv(POIo(t),t)] = -λ(Vons(PART))*∂[∂(t): E0pdv(POIo(t),t)] = -λ(Vons(PART))*Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[ sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) ] where : λ(Vons(PART)) is a positive constant Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 /*But this is not used directly in this "proper vector approach", for which a rederivation is necessary to be sure!!! /%From "EI_LENZodv(POIo(t),t) = EI_LENZpdv(POIo(t),t), using Lenz's Induction Law" : (2)** EIodv(POIo) = EI_LENZpdv(POIo(t),t) = -λ(Vons(PART))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) Subbing (2)** into definition of ∂[∂(t): EI_LENZods(POIo(t),t)] : (1) ∂[∂(t): EI_LENZods(POIo(t),t)] = ∂[∂(t): |EI_LENZodv(POIo(t),t)|] = ∂[∂(t): | - λ(Vons(PART))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t)|] (1) = λ(Vons(PART))*|Q(PART)|*∂[∂(t): |Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2|] From "∂[∂(t): E0ods(POIo,t)] = ∂[∂(t): E0pds(POIo(t),t)] - proper E0odv(POIo,t) vector approach" : /* Key intermediate result useful elsewhere : /% (12)* ∂[∂(t): |Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2|] = 2*cos(Aθpc(POIo(t),t))*Vons(PART)/Rpcs(POIo(t),t)^3 /*Subbing (12)* into (1) : /% (1) ∂[∂(t): EI_LENZods(POIo(t),t)] = ∂[∂(t): |EI_LENZodv(POIo(t),t)|] = λ(Vons(PART))*|Q(PART)|*∂[∂(t): |Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2|] = λ(Vons(PART))*|Q(PART)|*2*cos(Aθpc(POIo(t),t))*Vons(PART)/Rpcs(POIo(t),t)^3 /*For now - ARBITRARY sign on equations for scalar derivatives, as I have to go back & fix the derivations (not as straightforward as it sounds!). /% ∂[∂(t): EI_LENZods(POIo(t),t)] has the opposite sign of ∂[∂(t): E0ods(POIo,t)] /*Note that for : /% -PI/2 <= Aθpc(POIo(t),t)) <= PI/2 : then cos(Aθpc(POIo(t),t)) >0 and ∂[∂(t): E0pds(POIo(t),t)] > 0, and ∂[∂(t): EI_LENZpds(POIo(t),t)] > 0 /*19Dec2017 The following is nonsensical, as PI/2 >= -PI/2 !!?? /% PI/2 <= Aθpc(POIo(t),t)) <= -PI/2 : then cos(Aθpc(POIo(t),t)) <0 and ∂[∂(t): E0pds(POIo(t),t)] > 0, and ∂[∂(t): EI_LENZpds(POIo(t),t)] > 0 /*For Vonv(PART) > 0 (as per definition of BOTH coordinate frames of reference!), these results make sense Summarizing : /% (mathH) ∂[∂(t): EI_LENZpds(POIo(t),t)] = 2*λ(Vons(PART))*|Q(PART)|*cos(Aθpc(POIo(t),t))*Vons(PART)/Rpcs(POIo(t),t)^3 (endMath) /*+-----+ (RFo) basis /%From "Rpcs(POIo(t),t)" : (3)* Rpcs(POIo(t),t) = |Rpcv(POIo(t),t)| = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) From "cos(Aθpc(POIo(t),t))" : (1)* cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) Subbing (3)* & (1)* into (2) : (2) ∂[∂(t): EI_LENZods(POIo(t),t)] = 2*λ(Vons(PART))*|Q(PART)|*cos(Aθpc(POIo(t),t))*Vons(PART)/Rpcs(POIo(t),t)^3 = 2*λ(Vons(PART))*|Q(PART)|*Vons(PART) *cos(Aθpc(POIo(t),t)) /Rpcs(POIo(t),t)^3 = 2*λ(Vons(PART))*|Q(PART)|*Vons(PART) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^3 = 2*λ(Vons(PART))*|Q(PART)|*Vons(PART) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^2 Summarizing : (mathH) ∂[∂(t): EI_LENZods(POIo(t),t)] = 2*λ(Vons(PART))*|Q(PART)|*Vons(PART) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^2 (endMath) /********************* /%>>>>>>>>> ∂[∂(t): EI_LENZods(POIo(t),t)] = ∂[∂(t): EI_LENZpds(POIo(t),t)] - using Lenz's Law, cheating non-vector derivative of E0ods(POIo,t) /*+-----+ (RFp) basis From "Lenz's Induction Law and it's context" : /% (6)* ∂[∂(t): EI_LENZods(POIo(t),t)] = λ(Vons(PART))*∂[∂(t): E0ods(POIo)] /*+--+ From "E0ods(POIo,t) = E0pds(POIo(t),t)" : /% (1)* E0ods(POIo,t) = Q(PART)/Rpcs(POIp)^2 /*Taking the "cheating direct non-vector derivative" of (1)* : /% ∂[∂(t): E0ods(POIo,t)] = ∂[∂(t): Q(PART)/Rpcs(POIp)^2] = Q(PART)*∂[∂(t): 1/Rpcs(POIp)^2] (1) = Q(PART)*-2*Rpcs(POIp)^(-3)*∂[∂(t): Rpcs(POIp)] From "∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|]" : (1)** ∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|] = -Vons(PART)*cos(Aθpc(POIo(t),t)) Subbing (1)** into (1) : ∂[∂(t): E0ods(POIo,t)] = Q(PART)*-2*Rpcs(POIp)^(-3)*∂[∂(t): Rpcs(POIp)] = Q(PART)*-2*Rpcs(POIp)^(-3)*-Vons(PART)*cos(Aθpc(POIo(t),t)) (2) = 2*Q(PART)*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIp)^3 Subbing (2) into (6)* : (mathH) ∂[∂(t): E0ods(POIo,t)] = 2*λ(Vons(PART))*Q(PART)*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIp)^3 (endMath) /*+-----+ LIMIT CHECKS : From "Dimensional analysis (Gaussian units)" in file "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt" : E units = (charge/length^2) therefore ∂[∂(t): E] -> (charge/length^2/time) Dimensional consistency - OK, as RHS & LHS reduce to (charge/length^2/time). /%Comparing current result (2) to "∂[∂(t): EI_LENZods(POIo(t),t)] = ∂[∂(t): EI_LENZpds(POIo(t),t)] - using Lenz's Law, based on proper vector approach" (2)* : (2) ∂[∂(t): E0ods(POIo,t)] = -2*λ(Vons(PART))* Q(PART) *Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIp)^3 (2)* ∂[∂(t): EI_LENZods(POIo(t),t)] = 2*λ(Vons(PART))*|Q(PART)|*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 /*OTHER THAN the minus sign and |Q| (which should be obvious anyways), this is OK, so a "proper vector approach" was NOT actually necessary, but it is helpful for error trapping. /*+-----+ (RFo) basis Because (3) above is the same as for the "proper vector approach", there is no need to repeat exactly the same derivation here. /********************* >>>>>>>>> ∂[∂(t): ET_LENZodv(POIo(t),t)] = ∂[∂(t): ET_LENZpdv(POIo(t),t)], using Lenz's Induction Law /*+-----+ (RFp) basis /%From "ET_LENZodv(POIo(t),t) = ET_LENZpdv(POIo(t),t), using Lenz's Induction Law" : (1)* ET_LENZpdv(POIo(t),t) = (1 - λ(Vons(PART)))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) So : ∂[∂(t): ET_LENZpdv(POIo(t),t)] = ∂[∂(t): (1 - λ(Vons(PART)))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t)] For Chapter 4, [(1 - λ(Vons(PART))),Q(PART)] are constants with respect to (wrt) time. So : .1) ∂[∂(t): ET_LENZpdv(POIo(t),t)] = (1 - λ(Vons(PART)))*Q(PART) *∂[∂(t): Rpch(POIo(t),t)/Rpcs(POIo(t),t)^2] = (1 - λ(Vons(PART)))*Q(PART) *{ ∂[∂(t): Rpch(POIo(t),t)] / Rpcs(POIo(t),t)^2 + Rpch(POIo(t),t) *∂[∂(t): Rpcs(POIo(t),t)^( - 2)] } From "∂[∂(t): Rpch(POIo(t),t)]" : (2)* ∂[∂(t): Rpch(POIo(t),t)] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*RDEpdh(POIo(t),t) 22Jan2017 that's NOT the expression? where : λ(Vons(PART)) is a positive constant Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),t) is anchored at end of Rpch(POIo(t),t) and is at angle Aθpc(POIo(t),t) + PI/2, ie perpendicular to Rpch(POIo(t),t) angle Aφpc(POIo(t),t) doesn't change Looking at ∂[∂(t): Rpcs(POIo(t),t)^( - 2)] : (2) ∂[∂(t): Rpcs(POIo(t),t)^( - 2)] = (-2)*Rpcs(POIo(t),t)^(-3)*∂[∂(t): Rpcs(POIo(t),t)] From "∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|]" : (1)** ∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|] = -Vons(PART)*cos(Aθpc(POIo(t),t)) Subbing (1)** into (2) : (3) ∂[∂(t): Rpcs(POIo(t),t)^( - 2)] = (-2)*Rpcs(POIo(t),t)^(-3)*∂[∂(t): Rpcs(POIo(t),t)] = (-2)*Rpcs(POIo(t),t)^(-3)*-Vons(PART)*cos(Aθpc(POIo(t),t)) = 2*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 Subbing (2)* & (3) into (1) : (1) ∂[∂(t): ET_LENZpdv(POIo(t),t)] = (1 - λ(Vons(PART)))*Q(PART) *{ ∂[∂(t): Rpch(POIo(t),t)] / Rpcs(POIo(t),t)^2 + Rpch(POIo(t),t) *∂[∂(t): Rpcs(POIo(t),t)^( - 2)] } = (1 - λ(Vons(PART)))*Q(PART) *{ Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) *RDEpdh(POIo(t),t) / Rpcs(POIo(t),t)^2 + 2*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 *Rpch(POIo(t),t) } = (1 - λ(Vons(PART)))*Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *{ sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),t) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) } Summarizing : (mathH)/* Note that RDEpdh(POIo(t),t) & Rpch(POIo(t),t) are NOT the same unit vector! /% ∂[∂(t): ET_LENZpdv(POIo(t),t)] = (1 - λ(Vons(PART)))*Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *{sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),t) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t)} (endMath) /*+-----+ LIMIT CHECKS : From "Dimensional analysis (Gaussian units)" in file "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt" : ∂[∂(t): E] units = charge/length^2/time /%(4) ∂[∂(t): ET_LENZpdv(POIo(t),t)] (charge/length^2/time) = (1 - λ(Vons(PART)))*Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 charge length/time length^(-3) *{ sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),t) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) } /* (all dimensionless) = (charge/length^2/time) Dimensional consistency - OK, as RHS & LHS reduce to (charge/length^2/time). /%Equation (4) is (1 - λ(Vons(PART))) times "∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)]" Equation (6), as expected. /*+-----+ (RFo) basis (re-starting equation numbers) /%From "ET_LENZodv(POIo(t),t) = ET_LENZpdv(POIo(t),t), using Lenz's Induction Law" : (2)* ET_LENZodv(POIo(t),t) = (1 - λ(Vons(PART)))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } So : ∂[∂(t): ET_LENZodv(POIo(t),t)] = ∂[∂(t): (1 - λ(Vons(PART)))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } ] For Chapter 4, [(1 - λ(Vons(PART))),Q(PART)] are constants with respect to (wrt) time. = (1 - λ(Vons(PART)))*Q(PART) *∂[∂(t): Rpch(POIo(t),t)/{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}] (1) = (1 - λ(Vons(PART)))*Q(PART) *{ ∂[∂(t): Rpch(POIo(t),t)] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } + Rpch(POIo(t),t)*∂[∂(t): 1/{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}] } From "∂[∂(t): Rpch(POIo(t),t)]" : (2)* ∂[∂(t): Rpch(POIp)] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*RDEpdh(POIo(t),t) where RDEpdh(POIo(t),t) is anchored at end of Rpch(POIo(t),t) and is at angle Aθpc(POIo(t),t) + PI/2, ie perpendicular to Rpch(POIo(t),t) angle Aφpc(POIo(t),t) doesn't change Looking at : ∂[∂(t): 1/{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}] = (-1)*{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-2) *∂[∂(t): Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2] For Chapter 4, [Rocs(POIo),cos(Aθoc(POIo)),Vons(PART)] are constants with respect to (wrt) time. = (-1)*{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-2) *{ - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART) + Vons(PART)^2*2*t} = (-1)*{ - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART) + Vons(PART)^2*2*t } *{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-2) (2) = (-2)*{ - Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART) + Vons(PART)^2*t } /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 Subbing (2)* & (2) into (1) : (1) ∂[∂(t): ET_LENZodv(POIo(t),t)] = (1 - λ(Vons(PART)))*Q(PART) *{ ∂[∂(t): Rpch(POIo(t),t)] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } + Rpch(POIo(t),t)*∂[∂(t): 1/{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}] } = (1 - λ(Vons(PART)))*Q(PART) *{ Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*RDEpdh(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } + Rpch(POIo(t),t) * (-2)*{ - Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART) + Vons(PART)^2*t } /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 } (3) = (1 - λ(Vons(PART)))*Q(PART) *{ Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*RDEpdh(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } - 2*Vons(PART)*Rpch(POIo(t),t) *{ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t } /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART) *t]^2 }^2 } From "Rpcs(POIo(t),t)" : (3)* Rpcs(POIo(t),t) = |Rpcv(POIo(t),t)| = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) from "sin(Aθpc(POIo(t),t))" : (5)* sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (3)* & (5)* into (3) : (3) ∂[∂(t): ET_LENZodv(POIo(t),t)] = (1 - λ(Vons(PART)))*Q(PART) *{ Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*RDEpdh(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } - 2*Vons(PART)*Rpch(POIo(t),t) *{ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t } /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART) *t]^2 }^2 } = (1 - λ(Vons(PART)))*Q(PART) *{ Vons(PART)*RDEpdh(POIo(t),t) *sin(Aθpc(POIo(t),t)) /Rpcs(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } - 2*Vons(PART)*Rpch(POIo(t),t) *{ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t } /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART) *t]^2 }^2 } = (1 - λ(Vons(PART)))*Q(PART)*Vons(PART) *{ RDEpdh(POIo(t),t) * Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } - 2*Rpch(POIo(t),t) *{ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t } /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART) *t]^2 }^2 } = (1 - λ(Vons(PART)))*Q(PART)*Vons(PART) *{ Rocs(POIo)*sin(Aθoc(POIo)) *RDEpdh(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 - 2 *Rpch(POIo(t),t) *{ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t } /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART) *t]^2 }^2 } = (1 - λ(Vons(PART)))*Q(PART)*Vons(PART) *{ Rocs(POIo)*sin(Aθoc(POIo)) *RDEpdh(POIo(t),t) - 2*{ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t } *Rpch(POIo(t),t) } / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART) *t]^2 }^2 Summarizing : (mathH)/* where RDEpdh(POIo(t),t) is anchored at end of Rpch(POIo(t),t) and is at angle Aθpc(POIo(t),t) + PI/2, ie perpendicular to Rpch(POIo(t),t), angle Aφpc(POIo(t),t) doesn't change /% ∂[∂(t): ET_LENZodv(POIo(t),t)] = (1 - λ(Vons(PART)))*Q(PART)*Vons(PART) * { Rocs(POIo)*sin(Aθoc(POIo)) *RDEpdh(POIo(t),t) - 2*{ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t} *Rpch(POIo(t),t) } / {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo)) *Vons(PART)*t + [Vons(PART)*t]^2}^2 (endMath) /*+-----+ LIMIT CHECKS : From "Dimensional analysis (Gaussian units)" in file "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt" : ∂[∂(t): E] units = charge/length^2/time Dimensional consistency - OK, as RHS & LHS reduce to (charge/length^2/time). (just a quick eyeball look...) /%Equation (5) is (1 - λ(Vons(PART))) times "∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)]" Equation (9), as expected. /********************* >>>>>>>>> ∂[∂(t): BT_LENZodv(POIo(t),t)] = ∂[∂(t): BT_LENZpdv(POIo(t),t)] using Lenz's Induction Law, cheating scalar substitutions /%I will simplify the results of "∂[∂(t): BT_LENZodv(POIo(t),t)] = ∂[∂(t): BT_LENZpdv(POIo(t),t)] without use of Lenz's Induction Law" by substituting for EI_LENZods(POIo(t),t) & ∂[∂(t): EI_LENZods(POIo(t),t)] through the use of Lenz's Law. /*+-----+ (RFp) basis /%From "∂[∂(t): BT_LENZodv(POIo(t),t)] = ∂[∂(t): BT_LENZpdv(POIo(t),t)] without use of Lenz's Induction Law" : (6)* ∂[∂(t): BT_LENZpdv(POIo(t),t)] = ∂[∂(t): BT_LENZodv(POIo(t),t)] = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*Q(PART) /Rpcs(POIo(t),t)^3 - EI_LENZpds(POIo(t),t)/Rpcs(POIo(t),t) - ∂[∂(t): EI_LENZpds(POIo(t),t)]/Vons(PART)/cos(Aθpc(POIo(t),t)) } Improper derivative of Rodh(Vonv_X_Rpcv(POIo(t),t)) ] -> should use Rpdv(Vonv_X_Rpcv(POIo(t),t))/|Rpdv(Vonv_X_Rpcv(POIo(t),t))| with Kahan's formulation. (see From "EI_LENZods(POIo(t),t) = EI_LENZpds(POIo(t),t), using Lenz's Induction Law" : (2)* EI_LENZpds(POIo(t),t) = λ(Vons(PART))*|Q(PART)|/Rpcs(POIo(t),t)^2 From "∂[∂(t): EI_LENZods(POIo(t),t)] = ∂[∂(t): EI_LENZpds(POIo(t),t)] - using Lenz's Law, based on proper vector approach" : (2)** ∂[∂(t): EI_LENZods(POIo(t),t)] = 2*λ(Vons(PART))*|Q(PART)|*cos(Aθpc(POIo(t),t))*Vons(PART)/Rpcs(POIo(t),t)^3 Subbing (2)* & (2)** into (6)* : (6)* ∂[∂(t): BT_LENZpdv(POIo(t),t)] = ∂[∂(t): BT_LENZodv(POIo(t),t)] = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*Q(PART) /Rpcs(POIo(t),t)^3 - EI_LENZpds(POIo(t),t)/Rpcs(POIo(t),t) - ∂[∂(t): EI_LENZpds(POIo(t),t)]/Vons(PART)/cos(Aθpc(POIo(t),t)) } = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3 *Q(PART) /Rpcs(POIo(t),t)^3 - λ(Vons(PART))*|Q(PART)| /Rpcs(POIo(t),t)^2 /Rpcs(POIo(t),t) - 2*λ(Vons(PART))*|Q(PART)|*cos(Aθpc(POIo(t),t))*Vons(PART)/Rpcs(POIo(t),t)^3 /Vons(PART)/cos(Aθpc(POIo(t),t)) } = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3 *Q(PART) /Rpcs(POIo(t),t)^3 - λ(Vons(PART))*|Q(PART)|/Rpcs(POIo(t),t)^3 - 2*λ(Vons(PART))*|Q(PART)|/Rpcs(POIo(t),t)^3 } = Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *|Q(PART)|/Rpcs(POIo(t),t)^3 *{ 3 - 3*λ(Vons(PART)) } = 3*(1 - λ(Vons(PART)))*|Q(PART)|*Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 *Rodh(Vonv_X_Rpcv(POIo(t),t)) Summarizing : (mathH) ∂[∂(t): BT_LENZpdv(POIo(t),t)] = 3*(1 - λ(Vons(PART)))*|Q(PART)|*Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 *Rodh(Vonv_X_Rpcv(POIo(t),t)) (endMath) /*+--+ Limit CHECKS Dimensional consistency : From file "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt" sub-sub-section "Dimensional analysis (Gaussian units)" : As a convention here take [unit vectors, c] as dimensionless . ∂[∂(t): B] units = charge/length/time^2 /% (2) ∂[∂(t): BT_LENZpdv(POIo(t),t)] = ∂[∂(t): BT_LENZodv(POIo(t),t)] /* (charge/length/time^2) /% = 3*(1 + λ(Vons(PART)))*Q(PART)*Vons(PART)^2/c*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 *Rodh(Vonv_X_Rpcv(POIo(t),t)) = charge *(length/time)^2 /length^3 = (charge/length/time^2) /*OK, as all terms reduce to (dimensionless). Note : I have not yet re-derived "∂[∂(t): BT_LENZodv(POIo(t),t)] = ∂[∂(t): BT_LENZpdv(POIo(t),t)] using Lenz's Induction Law" using a proper vector approach. /*+-----+ (RFo) basis /%From "∂[∂(t): BT_LENZodv(POIo(t),t)] = ∂[∂(t): BT_LENZpdv(POIo(t),t)] without use of Lenz's Induction Law" : (6)* ∂[∂(t): BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t)] = Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t))* { - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(3/2) *[ Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } - EI_LENZods(POIo(t),t) ] + sin(Aθpc(POIo(t),t)) *[ Q(PART)*-2*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 - ∂[∂(t): EI_LENZods(POIo(t),t)] ] } /*Note that BOTH [ EI_LENZodv(POIo(t),t), ∂[∂(t): EI_LENZodv(POIo(t),t)] ] have changing direction & magnitude, and are NOT collinear with the (constant magnitude & direction) unit vector, Rodh(Vonv_X_Rpcv(POIo)). Therefore, I am concerned that direct substitution for [ EI_LENZods(POIo(t),t), ∂[∂(t): EI_LENZods(POIo(t),t)] ] into (6)* will NOT be appropriate, and that a more detailed "vector calculus" check is needed!! However, for now I proceed with direct substitution... /%From "EI_LENZods(POIo(t),t) = EI_LENZpds(POIo(t),t), using Lenz's Induction Law" : (3)* EI_LENZods(POIo(t),t) = EI_LENZpds(POIo(t),t) = λ(Vons(PART))*|Q(PART)| /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } From "∂[∂(t): EI_LENZods(POIo(t),t)] = ∂[∂(t): EI_LENZpds(POIo(t),t)] - using Lenz's Law, based on proper vector approach" : (3)** ∂[∂(t): EI_LENZods(POIo(t),t)] = 2*λ(Vons(PART))*|Q(PART)|*Vons(PART) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^2 Substituting (3)*,(3)** into (6)* : ∂[∂(t): BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t)] = Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t))* { - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(3/2) *[ |Q(PART)| /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } - EI_LENZods(POIo(t),t) ] + sin(Aθpc(POIo(t),t)) *[ |Q(PART)|*-2*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 - ∂[∂(t): EI_LENZods(POIo(t),t)] ] } = Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t))* { - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(3/2) *[ |Q(PART)| /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } - λ(Vons(PART))*|Q(PART)| /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } ] + sin(Aθpc(POIo(t),t)) *[ -2*|Q(PART)|*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 - 2*λ(Vons(PART))*|Q(PART)|*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^2 ] } = Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t))* { - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(3/2) *[ |Q(PART)| /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } * (1 - λ(Vons(PART))) ] + sin(Aθpc(POIo(t),t)) *[ -2*|Q(PART)|*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 * (1 - λ(Vons(PART))) ] } (1) = (1 - λ(Vons(PART)))*|Q(PART)|*Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(3/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } ] + sin(Aθpc(POIo(t),t)) *[ -2*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 ] } from "sin(Aθpc(POIo(t),t))" : (5)* sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (5)* into (1) : (1) ∂[∂(t): BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t)] = (1 - λ(Vons(PART)))*|Q(PART)|*Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(3/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } ] + Rocs(POIo)*sin(Aθoc(POIo)) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) *[ -2*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 ] } = (1 - λ(Vons(PART)))*|Q(PART)|*Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(5/2) ] + Rocs(POIo)*sin(Aθoc(POIo)) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) * -2*Vons(PART) * [ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] ] } = (1 - λ(Vons(PART)))*|Q(PART)|*Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) * [ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) *{ - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) + Rocs(POIo)*sin(Aθoc(POIo))*-2*Vons(PART) } = (1 - λ(Vons(PART)))*|Q(PART)|*Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) * [ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) *{ - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) + - 2*Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) } = (1 - λ(Vons(PART)))*|Q(PART)|*Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) * [ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) * - 3*Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) = 3*(1 - λ(Vons(PART)))*|Q(PART)|*Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) * Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART)* [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) = 3*(1 - λ(Vons(PART)))*|Q(PART)|*Vons(PART)^2/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) * Rocs(POIo)*sin(Aθoc(POIo))* [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) Summarizing : (mathH)/* !!!Note : problem with |Q(PART)| versus Q(PART) ignored here!!! /% BT_LENZodv(POIo(t),t)] = 3*(1 - λ(Vons(PART)))*|Q(PART)|*Vons(PART)^2/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) * Rocs(POIo)*sin(Aθoc(POIo))* [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) (endMath)\ /*+--+ Limit CHECKS Dimensional consistency : From file "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt" sub-sub-section "Dimensional analysis (Gaussian units)" : As a convention here take [unit vectors, c] as dimensionless . B units = (charge/length/time) (ignoring permittivity & permeability for Gaussian units) ∂[∂(t): B] units = charge/length/time^2 Showing units for (3) : /% ∂[∂(t): BT_LENZpdv(POIo(t),t) charge/length/time^2 = (1 + λ(Vons(PART)))*Q(PART)*Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) (charge) (length/time)^2 *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] [ (length) + (length/time)*time ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 /{ (length)^2 - length*(length/time)*time + [(length/time)*time]^2 }^2 *{ - Rocs(POIo) length /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(1/2) /* /[ (length)^2 - length*(length/time)*time + [(length/time)*time]^2]^(1/2) + -2 } = (charge) * (length/time)^2 * [ (length) + (length/time)*time ] /{ (length)^2 - length*(length/time)*time + [(length/time)*time]^2 }^2 *{ length /[ (length)^2 - length*(length/time)*time + [(length/time)*time]^2]^(1/2) } = charge*(length/time)^2 * (length) /(length)^2^2 * length /(length)^2^(1/2) = charge*(length/time)^2 * (length) /(length)^4 * length /(length) = (charge*length^(4-5)/time^2) = (charge*length/time^2) OK, as all terms reduce to (charge/length/time^2). /*+-----+ Rederiving the (RFp) basis - as a check As is often the case, more compact expressions can result from mixing (RFp) & (RFo) terms. Repeating : /% (3) ∂[∂(t): BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t)] = (1 + λ(Vons(PART)))*Q(PART)*Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 *{ - Rocs(POIo) /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(1/2) + -2 } /*!!!Note : problem with |Q(PART)| versus Q(PART) ignored here!!! /%From "cos(Aθpc(POIo(t),t))" : (1)* cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) From "sin(Aθpc(POIo(t),t))" : (5)* sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (1)* & (5)* into (3) : (3) ∂[∂(t): BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t)] = (1 + λ(Vons(PART)))*Q(PART)*Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *cos(Aθpc(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(2-1/2) *{ - Rocs(POIo) /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(1/2) + -2 } = (1 + λ(Vons(PART)))*Q(PART)*Vons(PART)^2/c*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) *{ - Rocs(POIo)*sin(Aθpc(POIo(t),t)) /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(1/2) + -2*sin(Aθpc(POIo(t),t)) } = (1 + λ(Vons(PART)))*Q(PART)*Vons(PART)^2/c*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) *{ - sin(Aθpc(POIo(t),t)) + -2*sin(Aθpc(POIo(t),t)) } = (1 + λ(Vons(PART)))*Q(PART)*Vons(PART)^2/c*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) * -3*sin(Aθpc(POIo(t),t)) (4) = -3*(1 + λ(Vons(PART)))*Q(PART)*Vons(PART)^2/c *sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) From "Rpcs(POIo(t),t)" : (3) Rpcs(POIo(t),t) = |Rpcv(POIo(t),t)| = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (3)* into (4) ∂[∂(t): BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t)] = -3*(1 + λ(Vons(PART)))*Q(PART)*Vons(PART)^2/c *sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) = -3*(1 + λ(Vons(PART)))*Q(PART)*Vons(PART)^2/c *sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) Summarizing : (5) ∂[∂(t): BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t)] = -3*(1 + λ(Vons(PART)))*Q(PART)*Vons(PART)^2/c*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 *Rodh(Vonv_X_Rpcv(POIo(t),t)) (RFp) basis - comparing to (2) above : (2) ∂[∂(t): BT_LENZpdv(POIo(t),t)] = ∂[∂(t): BT_LENZodv(POIo(t),t)] = 3*(1 + λ(Vons(PART)))*Q(PART)*Vons(PART)^2/c*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 *Rodh(Vonv_X_Rpcv(POIo(t),t)) /*+--+ Limit CHECKS AWESOME! The ONLY difference is the minus sign, but I'll have to better handle vector norms (as mentioned above!). While this "ring around the rosie" verification doesn't prove that either (2) or (3) are correct, it does give confidence at least in the mechanics of the derivation of (3) from (2) (closed loop calcs). /********************************************** >>>>>> Basics and Calculus using Lucas's results from Thomas Barnes iterations /********************* >>>>>>>>> Thomas Barnes iterations, it's context and relation to Lenz's Induction Law Expressions for EOodv(POIo(t),t) and its derivatives can be derived from simple geometry and basic laws. Lenz's law can be applied by using an unknown function of velocity, lambda(v), in order to derive expressions for the INDUCED & TOTAL electric and magnetic fields. This is what I have done in the sub-section of this document "III.4 Basics and Calculus using Lenz's Induction Law". However, the way that I am using terminology here, is that in the section "", the phrase "using Lenz's Induction Law" means that only the "first effect" of the E field is accounted for. In the section "III.5 Basics and Calculus using Lucas's results from Thomas Barnes iterations", an infinite series of feedbacks is accounted for to estimate the full effect of the E field, and therefore to estimate the full (rather than "first pass") magnetic field as well. The "first pass" results in the sub-sub-sections "using Lenz's Induction Law", turns out to yield the simple expression for lambda(v) : (1) EI = lambda(v)*E0 , where lambda(v) = beta^2 = (v/c)^2 beta^2 = (v/c)^2 which is commonly known as beta. Beta is part of the relativistic factor that arises from Relativity Theory, but that is shown to be superfluous and more "grounded" in the Barnes approach. Lucas follows Thomas Barnes work in using an iterative calculation to derive the full expressions for E & B fields, starting with "first pass" estimates arising from "using Lenz's Induction Law". The full ET (total, iterated Electric field or magnetic field), that arises is described by : /% (mathH) ETodv(POIo,t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*E0odv (endMath) where f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/(1 - [β*sin(Aθoc(POIo))]^2)^(3/2) , and again β = Vonv(PART)/c (3) BI = Vonv(PART)/c X ETodv(POIo,t) where "X" denotes the vector cross-product I do the iterative calculation to derive f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) in the file "Howell - math of Lucas Universal Force.ndf", which is the main document that assesses Lucas's Chapter 4. /********************* >>>>>>>>> Howell's correction to the f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) iteration factor ???NO - this would be for ∂[∂(t): EI] !!!??? In the sub-sub-sections below, I have simply used Lucas's final expressions after deriving /% f_BARNES(β,lambda). /*These are compared to my own results from applying Lenz's Indiuction Law, using hte general & unknown lambda(v) function of speed. /********************* >>>>>>>>> ET_BARNodv(POIo(t),t) = ET_BARNpdv(POIo(t),t), using Lucas's results from Thomas Barnes iterations (needs verification!) /*+-----+ (RFp) basis /%NOTE: EI_LENZpdv(POIo(t),t) = EIodv(POIo,t) I am ASSUMING that E0odv(POIo,t) & EIodv(POIo,t) are in the same direction, Rpch(POIo(t),t)!!! /*From Lucas's book p73h0.8 Equation (4-43) : /%(4-43) ET_BARNpdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*E0pdv(POIo(t),t) where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2) β = Vons(PART)/c in the context of Chapter 4 From "E0odv(POIo,t) = E0pdv(POIo(t),t)" : (1)* E0pdv(POIo(t),t) = E0odv(POIo,t) = Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) Subbing (1)* into (4-43) : ET_BARNpdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*E0pdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)/Rpcs(POIp)^2*Rpch(POIo(t),t) (mathH)/* where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2), β = Vons(PART)/c = (λ(Vons(PART)))^(1/2) in the context of Chapter 4 /% ET_BARNpdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)/Rpcs(POIp)^2*Rpch(POIo(t),t) (endMath) /*+--+ CHECKS Dimensional consistency - OK, as all terms reduce to (charge/length^2). (ignoring electric permeability for Gaussian coordinates) Note that "beta = Vons(PART)/c" is dimensionless /*+-----+ (RFo) basis As the E field is a function of the POIo and the charged particle, but independent of the reference frame : /%(1) ET_BARNodv(POIo(t),t) = ET_BARNpdv(POIo(t),t) /*From (1) in "(RFp) basis" above : /% (1)* ET_BARNpdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)/Rpcs(POIp)^2*Rpch(POIo(t),t) where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2) β = Vons(PART)/c = (λ(Vons(PART)))^(1/2) in the context of Chapter 4 From "Rpcs(POIo(t),t)" : (3)* Rpcs(POIo(t),t) = |Rpcv(POIo(t),t)| = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (1)* & (3)* into (1) : ET_BARNodv(POIo(t),t) = ET_BARNpdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)/Rpcs(POIp)^2*Rpch(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Rpch(POIo(t),t) /Rpcs(POIp)^2 = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^2 Leaving [_BARNES(Vonv(PART),Aθpd(POIo(t),t)), Rpch(POIo(t),t)] in (RFp) coordiantes for convenience, and summarizing : (mathH)/* where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2), β = Vons(PART)/c = (λ(Vons(PART)))^(1/2) in the context of Chapter 4 /% ET_BARNodv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Rpch(POIo(t),t) / {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}^(1/2)^2 (endMath) /*+--+ CHECKS Dimensional consistency - OK, as all terms reduce to (charge/length^2). (ignoring electric permeability for Gaussian coordinates) Note that "beta = Vons(PART)/c" is dimensionless, as is the unit vector Rpch(POIo(t),t) /********************* /%>>>>>>>>> ET_BARNods(POIo(t),t) = ET_BARNpds(POIo(t),t), using Lucas's results from Thomas Barnes iterations /*See the file "Howell - math of Lucas Universal Force.ndf", equation (4-33) for the details of the iterations. Here I address the question of the relative DIRECTIONs of E0odv(POIo,t) & EIodv(POIo,t), and rexpress Lucas's result for EIpds(POIo(t),t). NOTE: EIpdv(POIo(t),t) = EIodv(POIo,t) I am ASSUMING that E0odv(POIo,t) & EIodv(POIo,t) are in the same direction!!! Note : Lenz's Law (Lucas p64h0.5 Eq (4-5), p70h0.9 Eq (4-31)) provides assurance that E0odv(POIo,t) & EIodv(POIo,t) are in the same direction ... ??? /*+-----+ Proof that E0odv(POIo,t) & EIodv(POIo,t) are in the same direction This needs to be done later, but at first glance, it seems reasonable to assume that this is the case, as per Lenz's induction law, which is Equation (4-6) p64h0.5 in Lucas's book. Later ..... a more formal proof must be done. /*+-----+ (RFp) basis By definition : /%(1) ET_BARNpds(POIo(t),t) = |ET_BARNpdv(POIo(t),t)| From "ET_BARNodv(POIo(t),t) = ET_BARNpdv(POIo(t),t), using Lucas's results from Thomas Barnes iterations" : (1)* ET_BARNpdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)/Rpcs(POIp)^2*Rpch(POIo(t),t) where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2) β = Vons(PART)/c = (λ(Vons(PART)))^(1/2) in the context of Chapter 4 Subbing (1)* into (1) : ET_BARNpds(POIo(t),t) = |ET_BARNpdv(POIo(t),t)| = |f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)/Rpcs(POIp)^2*Rpch(POIo(t),t)| As [f_BARNES(Vonv(PART),Aθpd(POIo(t),t)),|Q(PART)|,Rpcs(POIp)^2] are >=0, |Rpch(POIo(t),t)| = 1 (unit vector) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|/Rpcs(POIp)^2 Summarizing : (mathH)/* where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2), β = Vons(PART)/c = (λ(Vons(PART)))^(1/2) in the context of Chapter 4 /% ET_BARNpds(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|/Rpcs(POIp)^2 (endMath) /*+--+ CHECKS Dimensional consistency - OK, as all terms reduce to (charge/length^2). (ignoring electric permeability for Gaussian coordinates) Note that [f_BARNES(Vonv(PART),Aθpd(POIo(t),t)), lambda(v), beta] are all dimensionless /*+-----+ (RFo) basis Again, by definition : /%(1) ET_BARNods(POIo(t),t) = |ET_BARNodv(POIo(t),t)| NO LONGER : IMPORTANT!!! Here I have assumed that Lucas's use of sin(Aθod(POIo)) was really intended to be sin(Aθpd(POIo(t),t)), as appears in the result above. From "ET_BARNodv(POIo(t),t) = ET_BARNpdv(POIo(t),t), using Lucas's results from Thomas Barnes iterations" "(RFo) basis" : (2)* ET_BARNodv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2) β = Vons(PART)/c = (λ(Vons(PART)))^(1/2) in the context of Chapter 4 Subbing (2)* into (1) : ET_BARNods(POIo(t),t) = |ET_BARNodv(POIo(t),t)| = | f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } | As [ f_BARNES(Vonv(PART),Aθpd(POIo(t),t)),Rpcs(POIp)^2] are >=0, and |Rpch(POIo(t),t)| = 1 (unit vector) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)| /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } Summarizing : (mathH)/* where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2), β = Vons(PART)/c = (λ(Vons(PART)))^(1/2) in the context of Chapter 4 /% ET_BARNods(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)| /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2} (endMath) /*+--+ CHECKS Dimensional consistency : /%From 4) ET_BARNods(POIo(t),t) = Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } *(1 - β) /{1 - [ β*Rocs(POIo)*sin(Aθoc(POIo)) /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ]^2 }^(3/2) where : (2) β = Vons(PART)/c in the context of Chapter 4 Note that "β = Vons(PART)/c" is dimensionless, and expressing in units : = charge / length^2 *dimensionless /{1 - [ length/length ) ]^2}^(3/2) = charge/length^2 /*OK, as all terms reduce to (charge/length^2). (ignoring electric permeability for Gaussian coordinates) Note that "beta = Vons(PART)/c" is dimensionless /********************* >>>>>>>>> BT_BARNodv(POIo(t),t) = BT_BARNpdv(POIo(t),t), using Lucas's results based on Thomas Barnes iterations (needs verification!) /*+-----+ (RFp) basis As per Lucas p73h0.75 Equation (4-43) part 2 (ie 4-43b or something like that) : /%(4-43b) BI_BARNpdv(POIo(t),t) = Vonv(PART)/c X ET_BARNpdv(POIo(t),t) /*Under the assumption that the are NO other magnetic fields B from magnetic particles or other electric fields : /%(1) BT_BARNpdv(POIo(t),t) = BI_BARNpdv(POIo(t),t) From "ET_BARNodv(POIo(t),t) = ET_BARNpdv(POIo(t),t), using Lucas's results from Thomas Barnes iterations" : (1)* ET_BARNpdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)/Rpcs(POIp)^2*Rpch(POIo(t),t) where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2) β = Vons(PART)/c = (λ(Vons(PART)))^(1/2) in the context of Chapter 4 Subbing (1)* into (1) : BT_BARNpdv(POIo(t),t) = Vonv(PART)/c X ET_BARNpdv(POIo(t),t) (2) = Vonv(PART)/c X f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)/Rpcs(POIp)^2*Rpch(POIo(t),t) Considering the vector cross-product : Here the situation is as described in "BTodv(POIo,t) = BTpdv(POIo(t),t)" : (6) BTpdv(POIo(t),t) = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t) ] where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Aφpd(Vonv_X_Rpcv(POIo(t),t)) & Rodh(Vonv_X_Rpcv(POIo(t),t)) are CONSTANT for Chapter 4 See also numerous warnings and corrections in that sub-sub-section and others... Now BI_BARNpdv(POIo(t),t) IS BTpdv(POIo(t),t) for which f_BARNES provides a solution for EIpds(POIo(t),t) : [ Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t) ] = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)/Rpcs(POIp)^2 So here we can make the SAME statements for the direction (unit vector) for BI_BARNpdv(POIo(t),t) as we did for BTpdv(POIo(t),t). Re-stating (2) : BT_BARNpdv(POIo(t),t) = Vonv(PART)/c X f_BARNES(Vonv(PART),Aθpd(POIo(t),t))* Q(PART) /Rpcs(POIp)^2*Rpch(POIo(t),t) = Vons(PART)/c*sin(Aθpc(POIo(t),t))*f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|/Rpcs(POIp)^2*Rodh(Vonv_X_Rpcv(POIo(t),t)) Summarizing : (mathH)/* where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Aφpd(Vonv_X_Rpcv(POIo(t),t)) & Rodh(Vonv_X_Rpcv(POIo(t),t)) are CONSTANT for /% BTI_BARNpdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)/c*sin(Aθpc(POIo(t),t))/Rpcs(POIp)^2*Rodh(Vonv_X_Rpcv(POIo(t),t)) (endMath) /*+-----+ (RFo) basis Under the assumption that the are NO other magnetic fields B from magnetic particles or other electric fields : /%(1) BT_BARNodv(POIo(t),t) = BI_BARNodv(POIo(t),t) From "ET_BARNodv(POIo(t),t) = ET_BARNpdv(POIo(t),t), using Lucas's results from Thomas Barnes iterations" (RFo) basis : (2)* ET_BARNodv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2) β = Vons(PART)/c = (λ(Vons(PART)))^(1/2) in the context of Chapter 4 Subbing (2)* into (1) : BT_BARNodv(POIo(t),t) = Vonv(PART)/c X ET_BARNodv(POIo(t),t) (2) = Vonv(PART)/c X f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } Notice that we retain "Aθpd(POIo(t),t))" (RFp format) for convenience, but in any case it is a factor for f_BARNES, as opposed to being directly in the equation (2). Considering the vector cross-product : As with the RFo basis and as described in "BTodv(POIo,t) = BTpdv(POIo(t),t)" : (9) BTodv(POIo,t) = Vons(PART)/c *Rocs(POIo)*sin(Aθoc(POIo)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) - EIods(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ] where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) See also numerous warnings and corrections in that sub-sub-section and others... Now BI_BARNodv(POIo(t),t) IS BTodv(POIo,t) for which f_BARNES provides a solution for EIpds(POIo(t),t) : So here we can make the SAME statements for the direction (unit vector) for BI_BARNpdv(POIo(t),t) as we did for BTpdv(POIo(t),t). Re-stating (2), and retaining (RFp) terms [Rodh(Vonv_X_Rpcv(POIo(t),t)), sin(Aθpc(POIo(t),t))] for great convenience : BT_BARNodv(POIo(t),t) = Vonv(PART)/c X f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } = Vons(PART)/c*sin(Aθpc(POIo(t),t))*f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Rodh(Vonv_X_Rpcv(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } (3) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } From "sin(Aθpc(POIo(t),t))" : (5)* sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (5)* into (3) : BT_BARNodv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)/c *Rodh(Vonv_X_Rpcv(POIo(t),t)) *sin(Aθpc(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)/c *Rodh(Vonv_X_Rpcv(POIo(t),t)) * Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)/c*Rocs(POIo)*sin(Aθoc(POIo)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) Summarizing : (mathH)/* where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Aφpd(Vonv_X_Rpcv(POIo(t),t)) & Rodh(Vonv_X_Rpcv(POIo(t),t)) are CONSTANT for /% BT_BARNodv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)/c*Rocs(POIo)*sin(Aθoc(POIo)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}^(3/2) (endMath) /********************* /%>>>>>>>>> ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))], using Lucas's results based on Thomas Barnes iterations (needs verification!) /*+-----+ (RFp) FORMAT From Lucas's book p73h0.8 Equation (4-43), and my use of the symbol f_BARNES : /%(4-43) ET_BARNpdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*E0pdv(POIo(t),t) where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2) β = Vons(PART)/c in the context of Chapter 4 Therefore, by definition in (4-43) : (1) f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2) Note that f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) is a scalar (+ve,-ve), not a vector. With normal assumptions, f_BARNES >= 0, but I don't enforce that here. Taking the derivative of (1) : ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] = ∂[∂(t): (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2)] Given that [Vonv(PART), β] are constant in Chapter 4 : = (1 - β^2)*∂[∂(t): {1 - [β*sin(Aθpd(POIo(t),t))]^2}^( - 3/2)] = (1 - β^2)*(-3/2) *{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(-5/2) *∂[∂(t): 1 - [β*sin(Aθpd(POIo(t),t))]^2] = (1 - β^2)*(-3/2) *{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(-5/2) *(-1) *∂[∂(t): [β*sin(Aθpd(POIo(t),t))]^2] = (1 - β^2)*(-3/2) *{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(-5/2) *(-1)*2*β*sin(Aθpd(POIo(t),t)) *∂[∂(t): β*sin(Aθpd(POIo(t),t))] = (1 - β^2)*(-3/2) *{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(-5/2) *(-1)*2*β*sin(Aθpd(POIo(t),t))*β*∂[∂(t): sin(Aθpd(POIo(t),t))] (2) = 3*β^2*(1 - β^2)*{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(-5/2)*sin(Aθpd(POIo(t),t)) *∂[∂(t): sin(Aθpd(POIo(t),t))] From "∂[∂(t): sin(Aθpc(POIo(t),t))]" RFp basis : (2)* ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) Subbing (2)* into (2) : (2) ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] = 3*β^2*(1 - β^2)*{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(-5/2)*sin(Aθpd(POIo(t),t)) *∂[∂(t): sin(Aθpd(POIo(t),t))] = 3*β^2*(1 - β^2)*{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(-5/2)*sin(Aθpd(POIo(t),t)) *Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) (3) = 3*β^2*(1 - β^2)*{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(-5/2) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) Summarizing : (mathH) ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) (endMath) /*+-----+ (RFo) basis Converting RFp (3) above to RFo terms : (3)* ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] = 3*beta^2*(1 - beta^2)*{1 - [beta*sin(Aθpd(POIo(t),t))]^2}^(-5/2) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) From "sin(Aθpc(POIo(t),t))" RFo basis : (5)* sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) From "cos(Aθpc(POIo(t),t))" RFo basis: (1)* cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) From "Rpcs(POIo(t),t)" RFo basis : (3)** Rpcs(POIo(t),t) = |Rpcv(POIo(t),t)| = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) Subbing (5)*,(1)*,(3)** into (3)* (except for {1 - [beta*sin(Aθpd(POIo(t),t))]^2} term) : ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] = 3*beta^2*(1 - beta^2)*{1 - [beta*sin(Aθpd(POIo(t),t))]^2}^(-5/2) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) = 3*beta^2*(1 - beta^2)*{1 - [beta*sin(Aθpd(POIo(t),t))]^2}^(-5/2) *Vons(PART) *sin(Aθpc(POIo(t),t))^2 *cos(Aθpc(POIo(t),t)) /Rpcs(POIo(t),t) = 3*beta^2*(1 - beta^2)*{1 - [beta*sin(Aθpd(POIo(t),t))]^2}^(-5/2) *Vons(PART) *Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) (1) = 3*beta^2*(1 - beta^2)*{1 - [beta*sin(Aθpd(POIo(t),t))]^2}^(-5/2) *Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) Subbing for f_BARNES : = 3*beta^2/{1 - [beta*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) Summarizing : (mathH) ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] = 3*beta^2/{1 - [beta*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) *[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}^(3/2) (endMath) /********************* /%>>>>>>>>> ∂[∂(t): ET_BARNodv(POIo(t),t)] = ∂[∂(t): ET_BARNpdv(POIo(t),t)], using Lucas's results based on Thomas Barnes iterations (needs verification!) /*+-----+ (RFp) FORMAT From Lucas's book p73h0.8 Equation (4-43) : /%(4-43) ET_BARNpdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*E0pdv(POIo(t),t) where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2) β = Vons(PART)/c in the context of Chapter 4 From first principles : ∂[∂(t): ET_BARNpdv(POIo(t),t)] = ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*E0pdv(POIo(t),t)] (1) = ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))]*E0pdv(POIo(t),t) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*∂[∂(t): E0pdv(POIo(t),t)] /*+-----+ FIRST TERM - /%(2) ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))]*E0pdv(POIo(t),t) From "∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))], using Lucas's results based on Thomas Barnes iterations" : (4)* ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) From "E0odv(POIo,t) = E0pdv(POIo(t),t)" : (1)* E0pdv(POIo(t),t) = E0odv(POIo,t) = Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) Subbing (4)* & (1)* into (2) : ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))]*E0pdv(POIo(t),t) = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) *Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) (3) = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Q(PART)*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3*Rpch(POIo(t),t) /*+--+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (charge/length^2/time). (ignoring electric permeability for Gaussian coordinates) /% (3) ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))]*E0pdv(POIo(t),t) 1/time (charge/length^2) = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) dimensionless *Q(PART)*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3*Rpch(POIo(t),t) charge (length/time) /length^3 = charge/length^2/time /*+-----+ SECOND TERM - /%(4) f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*∂[∂(t): E0pdv(POIo(t),t)] From "∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)]" RFp basis : (6)* ∂[∂(t): E0pdv(POIo(t),t)] = Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[ sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) ] where : Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 Subbing (6)* into (4) : (5) f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*∂[∂(t): E0pdv(POIo(t),t)] = f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[ sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) ] where : Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 /*+--+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (charge/length^2/time). (ignoring electric permeability for Gaussian coordinates) /%(5) f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*∂[∂(t): E0pdv(POIo(t),t)] ∂[∂(t): E] units = charge/length^2/time = f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 charge * (length/time) /length^3 *[ sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) ] (no units this line) = (charge/length^2/time) /*+-----+ COMBINING TERMS Subbing (3) & (5) into (1) : /% (1) ∂[∂(t): ET_BARNpdv(POIo(t),t)] = ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] *E0pdv(POIo(t),t) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *∂[∂(t): E0pdv(POIo(t),t)] = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Q(PART)*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3*Rpch(POIo(t),t) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[ sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) ] = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 { 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} *sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) + *[ sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t) ] } = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[ Rpch(POIo(t),t) *{ 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t)) + 2*cos(Aθpc(POIo(t),t)) } + RDEpdh(POIo(t),∂(t)) *sin(Aθpc(POIo(t),t)) ] = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[ Rpch(POIo(t),t) *cos(Aθpc(POIo(t),t))*{ 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*sin(Aθpc(POIo(t),t))^2 + 2 } + RDEpdh(POIo(t),∂(t)) *sin(Aθpc(POIo(t),t)) ] Summarizing : (mathH)/* where : Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 /% ∂[∂(t): ET_BARNpdv(POIo(t),t)] = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[ Rpch(POIo(t),t) *cos(Aθpc(POIo(t),t))*{ 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*sin(Aθpc(POIo(t),t))^2 + 2 } + RDEpdh(POIo(t),∂(t)) *sin(Aθpc(POIo(t),t)) ] (endMath) 01Apr2016 - I don't trust this at all! /*+--+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (charge/length^2/time). (ignoring electric permeability for Gaussian coordinates) (same as for second term...) /*+-----+ SPECIAL CHECK As per the note above "This problematic in this derivation!!!!", this result must be checked to see if the original assumption regarding the use of differentiating ∂[∂(t): ET_BARNpds(POIo(t),t)]. Later ... /*+-----+ (RFo) basis : Following the same appraoch as for the RFp basis above From first principles : /% ∂[∂(t): ET_BARNodv(POIo(t),t)] = ∂[∂(t): ET_BARNpdv(POIo(t),t)] = ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*E0odv(POIo,t)] (1) = ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))]*E0odv(POIo,t) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*∂[∂(t): E0odv(POIo,t)] /*+-----+ FIRST TERM - /%(2) ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))]*E0odv(POIo,t) From "∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))], using Lucas's results based on Thomas Barnes iterations" : (2)* ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) From "E0odv(POIo,t) = E0pdv(POIo(t),t)" : (5)* E0odv(POIo) = E0pdv(POIo) = Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } where Rpch(POIo(t),t) = Rodh(POIo) = displacement vector [start : (POIo), length : 1, theta : arccos(Aθpc(POIo(t),t)), phi : Aφoc(POIo)] and cos(Aθpc(POIo(t),t)) = [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) Subbing (2)* & (5)* into (2) : (2) ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))]*E0odv(POIo,t) = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) *Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } (3) = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Q(PART)*Rpch(POIo(t),t)*Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) /*+-----+ SECOND TERM - /%(4) f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*∂[∂(t): E0odv(POIo,t)] From "∂[∂(t): E0odv(POIo,t)] = ∂[∂(t): E0pdv(POIo(t),t)]" RFp basis : (9) ∂[∂(t): E0pdv(POIo(t),t)] = Q(PART)*Vons(PART) *[ Rocs(POIo)*sin(Aθoc(POIo)) *RDEpdh(POIo(t),∂(t)) + 2*[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rpch(POIo(t),t) ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 where : Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 Aθpc(POIo(t),t) = arcsin [ Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ] = arccos [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^(1/2) ] Subbing (9)* into (4) (5) f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*∂[∂(t): E0odv(POIo,t)] = f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Q(PART)*Vons(PART) *[ Rocs(POIo)*sin(Aθoc(POIo)) *RDEpdh(POIo(t),∂(t)) + 2*[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rpch(POIo(t),t) ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 /*+-----+ COMBINING TERMS Subbing (3) & (5) into (1) : (1) ∂[∂(t): ET_BARNodv(POIo(t),t)] = ∂[∂(t): ET_BARNpdv(POIo(t),t)] = ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))]*E0odv(POIo,t) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*∂[∂(t): E0odv(POIo,t)] = 3*beta^2/{1 - [beta*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Q(PART)*Rpch(POIo(t),t)*Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Q(PART)*Vons(PART) *[ Rocs(POIo)*sin(Aθoc(POIo)) *RDEpdh(POIo(t),dt) + 2*[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rpch(POIo(t),t) ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 = Q(PART)*Vons(PART) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 *{ 3*beta^2/{1 - [beta*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Rpch(POIo(t),t)*Rocs(POIo)*sin(Aθoc(POIo)) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *[ Rocs(POIo)*sin(Aθoc(POIo)) *RDEpdh(POIo(t),dt) + 2*[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rpch(POIo(t),t) ] } = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Vons(PART) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 *{ Rpch(POIo(t),t) *[ 3*beta^2/{1 - [beta*sin(Aθpd(POIo(t),t))]^2} *Rocs(POIo)*sin(Aθoc(POIo)) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) + 2*[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] ] + RDEpdh(POIo(t),dt) *Rocs(POIo)*sin(Aθoc(POIo)) } = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Vons(PART) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 *{ Rpch(POIo(t),t) *[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *[ 3*beta^2/{1 - [beta*sin(Aθpd(POIo(t),t))]^2} * Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) + 2 ] + RDEpdh(POIo(t),dt) *Rocs(POIo)*sin(Aθoc(POIo)) } Summarizing : (mathH)/* where : Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),dt) is at angle Aθpd(RDEpdh(POIo(t),dt)) = Aθpc(POIo(t),t) + PI/2 /% ∂[∂(t): ET_BARNodv(POIo(t),t)] = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Vons(PART) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 *{ Rpch(POIo(t),t) *[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *[ 3*beta^2/{1 - [beta*sin(Aθpd(POIo(t),t))]^2} * Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) + 2 ] + RDEpdh(POIo(t),dt) *Rocs(POIo)*sin(Aθoc(POIo)) } (endMath) /*+--+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (charge/length^2/time). (ignoring electric permeability for Gaussian coordinates) /%(6) ∂[∂(t): ET_BARNodv(POIo(t),t)] (charge/length^2/time) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Vons(PART) charge * length/time / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 /length^4 *{ Rpch(POIo(t),t) *[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] length *[ 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} * Rocs(POIo)*sin(Aθoc(POIo)) length /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /length + 2 ] + RDEpdh(POIo(t),∂(t)) *Rocs(POIo)*sin(Aθoc(POIo)) length } /* = charge * length/time /length^4 *length = (charge/length^2/time) /********************* /%>>>>>>>>> ∂[∂(t): BTodv(POIo,t)] = ∂[∂(t): BTpdv(POIo(t),t)], using Lucas's results from Thomas Barnes iterations (needs verification!) Note that for any specific POIo, BTodv(POIo,t) = BTpdv(POIo(t),t) changes in magnitude and in [+,-] direction, but it DOES NOT change in direction of unit vector Rodh(Vonv_X_Rpcv(POIo(t),t)). [Aφpd(POIo(t),t), APod(POIo)] => direction of unit vector Rodh(Vonv_X_Rpcv(POIo(t),t)). where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Aφpd(Vonv_X_Rpcv(POIo(t),t)) & Rodh(Vonv_X_Rpcv(POIo(t),t)) are CONSTANTS for a given POIo Therefore, ∂[∂(t): BTodv(POIo,t)] = ∂[∂(t): BTpdv(POIo(t),t)] is always in direction of the unit vector Rodh(Vonv_X_Rpcv(POIo(t),t)) for a given POIo. It's magnitude channges, but not the direction (except [+,-]) of the unit vector Rodh(Vonv_X_Rpcv(POIo(t),t)). /*+-----+ (RFp) FORMAT /%From "BT_BARNodv(POIo(t),t) = BT_BARNpdv(POIo(t),t), using Lucas's results based on Thomas Barnes iterations" RFp basis : (3)* BT_BARNpdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)/c*sin(Aθpc(POIo(t),t))/Rpcs(POIp)^2*Rodh(Vonv_X_Rpcv(POIo(t),t)) where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Aφpd(Vonv_X_Rpcv(POIo(t),t)) & Rodh(Vonv_X_Rpcv(POIo(t),t)) are CONSTANT for Taking the derivative of (3)* : ∂[∂(t): BT_BARNpdv(POIo(t),t)] = ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)/c*sin(Aθpc(POIo(t),t))/Rpcs(POIp)^2*Rodh(Vonv_X_Rpcv(POIo(t),t))] As [|Q(PART)|,Vons(PART),c,Rodh(Vonv_X_Rpcv(POIo(t),t))] are constant with time (Rodh(Vonv_X_Rpcv(POIo(t),t)) always in same direction!!!) : = |Q(PART)|*Vons(PART)/c *Rodh(Vonv_X_Rpcv(POIo(t),t)) *∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*sin(Aθpc(POIo(t),t))/Rpcs(POIp)^2] (1) = |Q(PART)|*Vons(PART)/c *Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))]* sin(Aθpc(POIo(t),t)) *Rpcs(POIp)^(-2) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *∂[∂(t): sin(Aθpc(POIo(t),t))] *Rpcs(POIp)^(-2) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) * sin(Aθpc(POIo(t),t)) *∂[∂(t): Rpcs(POIp)^( - 2)] } From "∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))], using Lucas's results based on Thomas Barnes iterations" RFp basis : (4)* ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) From "∂[∂(t): sin(Aθpc(POIo(t),t))]" RFp basis : (2)* ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) From "∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|]" RFp basis : (1)* ∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|] = -Vons(PART)*cos(Aθpc(POIo(t),t)) Subbing (4)*,(2)*,(1)* into (1) : (1) ∂[∂(t): BT_BARNpdv(POIo(t),t)] = |Q(PART)|*Vons(PART)/c *Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))]* sin(Aθpc(POIo(t),t)) *Rpcs(POIp)^(-2) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *∂[∂(t): sin(Aθpc(POIo(t),t))] *Rpcs(POIp)^(-2) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) * sin(Aθpc(POIo(t),t)) *∂[∂(t): Rpcs(POIp)^( - 2)] } = |Q(PART)|*Vons(PART)/c *Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} *f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) *sin(Aθpc(POIo(t),t)) *Rpcs(POIp)^(-2) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Rpcs(POIp)^(-2) *Vons(PART)*sin(Aθpc(POIo(t),t)) *cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) * sin(Aθpc(POIo(t),t)) *(-2)*Rpcs(POIp)^(-3)*∂[∂(t): Rpcs(POIp)] } = |Q(PART)|*Vons(PART)/c *Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} *f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) *sin(Aθpc(POIo(t),t)) *Rpcs(POIp)^(-2) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Rpcs(POIp)^(-2) *Vons(PART)*sin(Aθpc(POIo(t),t)) *cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) * sin(Aθpc(POIo(t),t)) *(-2)*Rpcs(POIp)^(-3) *-Vons(PART)*cos(Aθpc(POIo(t),t)) } = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)^2/c*cos(Aθpc(POIo(t),t)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} *sin(Aθpc(POIo(t),t))^2/Rpcs(POIo(t),t) *sin(Aθpc(POIo(t),t)*Rpcs(POIp)^( - 2) + sin(Aθpc(POIo(t),t)) /Rpcs(POIo(t),t) *Rpcs(POIp)^(-2) + sin(Aθpc(POIo(t),t)) *(-2)*Rpcs(POIp)^(-3) *(-1) } = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)^2/c*cos(Aθpc(POIo(t),t)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ { 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} *sin(Aθpc(POIo(t),t))^3/Rpcs(POIo(t),t)^3 + sin(Aθpc(POIo(t),t)) /Rpcs(POIo(t),t)^3 + 2*sin(Aθpc(POIo(t),t)) /Rpcs(POIp)^3 } = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIp)^3 *Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ { 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} *sin(Aθpc(POIo(t),t))^2 + 1 + 2 } = 3*f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIp)^3 *{ { [β*sin(Aθpc(POIo(t),t))]^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} + 1 } *Rodh(Vonv_X_Rpcv(POIo(t),t)) Summarizing : (mathH)/* where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Aφpd(Vonv_X_Rpcv(POIo(t),t)) & Rodh(Vonv_X_Rpcv(POIo(t),t)) are CONSTANT for /% ∂[∂(t): BT_BARNpdv(POIo(t),t)] = 3*f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIp)^3 *{ { [β*sin(Aθpc(POIo(t),t))]^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} + 1} *Rodh(Vonv_X_Rpcv(POIo(t),t)) (endMath) /*+--+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (charge/length/time^2). (ignoring electric permeability for Gaussian coordinates) See file : "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt" sub-section : "Dimensional analysis (Gaussian units)" : /%(3) ∂[∂(t): BT_BARNodv(POIo(t),t)] ∂[∂(t): B] units = charge/length/time^2 = 3*f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIp)^3 charge (length/time)^2 /length^3 *{ { [β*sin(Aθpc(POIo(t),t))]^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} + 1 } *Rodh(Vonv_X_Rpcv(POIo(t),t)) dimensionless line ... /* = charge/length/time^2 /*+-----+ (RFo) basis : /%From "BT_BARNodv(POIo(t),t) = BT_BARNpdv(POIo(t),t), using Lucas's results based on Thomas Barnes iterations" RFo basis : (4)* BT_BARNodv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)/c*Rocs(POIo)*sin(Aθoc(POIo)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Aφpd(Vonv_X_Rpcv(POIo(t),t)) & Rodh(Vonv_X_Rpcv(POIo(t),t)) are CONSTANT for Taking the time derivative of (4)* : ∂[∂(t): BT_BARNodv(POIo(t),t)] = ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)/c*Rocs(POIo)*sin(Aθoc(POIo))*Rodh(Vonv_X_Rpcv(POIo(t),t)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) ] As [|Q(PART)|,Vons(PART),c,Rocs(POIo),sin(Aθoc(POIo)),cos(Aθoc(POIo)),Rodh(Vonv_X_Rpcv(POIo(t),t))] are constant at a given POIo : = |Q(PART)|*Vons(PART)/c*Rocs(POIo)*sin(Aθoc(POIo)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-3/2) ] (1) = |Q(PART)|*Vons(PART)/c*Rocs(POIo)*sin(Aθoc(POIo)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] *{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-3/2) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *∂[∂(t): {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}^( - 3/2)] } From "∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))], using Lucas's results based on Thomas Barnes iterations" RFo basis : (2)* ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) Looking at : ∂[∂(t): {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}^( - 3/2)] As [Rocs(POIo),sin(Aθoc(POIo)),cos(Aθoc(POIo)),Rodh(Vonv_X_Rpcv(POIo(t),t))] are constant at a given POIo : = ∂[∂(t): {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}^( - 3/2)] = (-3/2) *{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-5/2) ∂[∂(t): {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}] = (-3/2) *{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-5/2) *{ ∂[∂(t): Rocs(POIo)^2] - ∂[∂(t): 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + ∂[∂(t): [Vons(PART)*t]^2] } = (-3/2) *{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-5/2) *{ 0 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART) + 2*Vons(PART)*t*∂[∂(t): Vons(PART)*t] } = (-3/2) *{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-5/2) *{ - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART) + 2*Vons(PART)*t*Vons(PART) } = (-3/2)*2*Vons(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) *{ - Rocs(POIo)*cos(Aθoc(POIo)) + t*Vons(PART) } = 3*Vons(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) *{ + Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t } (2) = 3*Vons(PART) *{ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t } /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) Subbing (2)*,(2) into (1) : ∂[∂(t): BT_BARNodv(POIo(t),t)] = |Q(PART)|*Vons(PART)/c*Rocs(POIo)*sin(Aθoc(POIo)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) *{ 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) *{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(-3/2) + f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) * 3*Vons(PART) *{ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t } /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) } = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)^2/c*Rocs(POIo)*sin(Aθoc(POIo)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) *{ 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} *Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) + 3 } Summarizing : (mathH)/* where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Aφpd(Vonv_X_Rpcv(POIo(t),t)) & Rodh(Vonv_X_Rpcv(POIo(t),t)) are CONSTANT for /% ∂[∂(t): BT_BARNodv(POIo(t),t)] = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)^2/c*Rocs(POIo)*sin(Aθoc(POIo)) * [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rodh(Vonv_X_Rpcv(POIo(t),t)) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) *{ 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} *Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) + 3 } (endMath) /*+--+ LIMIT CHECKS : Dimensional consistency - OK, as all terms reduce to (charge/length/time^2). (ignoring electric permeability for Gaussian coordinates) See file : "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt" sub-section : "Dimensional analysis (Gaussian units) " : /%(3) ∂[∂(t): BT_BARNodv(POIo(t),t)] ∂[∂(t): B] units = charge/length/time^2 = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)^2/c*Rocs(POIo)*sin(Aθoc(POIo)) charge (length/time)^2 length * [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rodh(Vonv_X_Rpcv(POIo(t),t)) length / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) /length^5 *{ 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} *Rocs(POIo)*sin(Aθoc(POIo)) length /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) length + 3 } /* = charge * (length/time)^2 * length * length / length^5 = charge/length/time^2 endchapter **************************************** >>> IV. Expressions for the t=0 reference frame (RFt) In order to force myself to be "more correct and consistent" in switching "contexts or reference frames" in going between integrations and derivatives, I found it helpful to define new symbols and the "t=o reference frame" (RFt). 14Jun2016 As I have just adopted this approach, this chapter is woefully inadequate, and in need of much greater detail. Right now, the ONLY content are descriptions of : reference frame switches (RFo -> RFt) resulting from , and RFt -> RFo) and corresponding variable changes only one derivative as shown below. This needs a figure. /********************* >>>>>>>>> Procedures for consistent reference frame switching From below : Chapter "Derivatives & Integrals adapted to Chapter 4", subsection "Summary" Procedures - integrals /* For derivatives in next step : Actually, by incorporating this into the derivatives & integrals, most cases are handled "automatically" /* integration ∫{dAOtc, 0 to Aθpc(POIo(t)=0) : expressions with [Rocs(POIo),Rpcs(POIo(t)=0),AOtc(RFo ),AOtc(RFp) ,E0ods(POIo(t)=0)] } /* result gives [Rocs(POIo),Rpcs(POIo(t),t) ,Aθoc(POIo),Aθpc(POIo(t)=0),E0ods(POIo,t) ] } Procedures - derivatives /* For integrals in next step : Actually, by incorporating this into the derivatives & integrals, most cases are handled "automatically" /* derivative ∂[∂(t): expressions with [Rocs(POIo),Rpcs(POIo(t),t) ,Aθoc(POIo),Aθpc(POIo(t),t),E0ods(POIo,t) ] /* result gives [Rocs(POIo),Rpcs(POIo(t)=0),AOtc(RFo) ,AOtc(RFp) ,E0ods(POIo(t)=0)] /********************* >>>>>>>>> Calculus of RFt /********************* /%>>>>>>>>> ∂[∂(Aθpc): Rpcs(POIo(t),t=0)] = ∂[∂(Aθpc): |Rpcv(POIo(t),t)|] (is this wrong?) /*NOTE : This should be done by a proper vector derivative approach!! 14Jun2016 - Is this WRONG??? CONTEXT : Important -> In the equations of Chapter, we are INTEGRATING with respect to (wrt) Aθpc, rather than differentiating. Simple non-vector approach : /%From "∂[∂(t): Rpcv(POIo(t),t)]" 1*) ∂[∂(t): Rpcv(POIo(t),t)] = -Vonv(PART) From Figure "∂[∂(t): Rpcs(POIo(t),t)]" ROPI2pds(POIo(t),t) = ROPI2ods(POIo) = constant = Rocs(POIo) *sin(Aθoc(POIo)) = Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t)) Or : (1) Rpcs(POIo(t),t) = Rocs(POIo)*sin(Aθoc(POIo))/sin(Aθpc(POIo(t),t)) Differentiating with respect to (wrt) Aθpc : ∂[∂(Aθpc): Rpcs(POIo(t),t)] = ∂[∂(Aθpc): Rocs(POIo)*sin(Aθoc(POIo))/sin(Aθpc(POIo(t),t))] But Rocs(POIo),sin(Aθoc(POIo)) are constants : = Rocs(POIo)*sin(Aθoc(POIo))*∂[∂(Aθpc): sin(Aθpc(POIo(t),t)^( - 1)] = Rocs(POIo)*sin(Aθoc(POIo))*(-1)*sin(Aθpc(POIo(t),t))^(-2)*∂[∂(Aθpc): sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo))*(-1)*sin(Aθpc(POIo(t),t))^(-2)*cos(Aθpc(POIo(t),t) Summarizing : (mathH) ∂[∂(Aθpc): Rpcs(POIo(t),t)] = (-1)*Rocs(POIo)*sin(Aθoc(POIo))*sin(Aθpc(POIo(t),t))^(-2)*cos(Aθpc(POIo(t),t) (endMath) /************************************************* >>> APPENDICES /************************************************* >>>>>> Future extensions of the Universal Force /*$ cat >>"$p_augmented" "$d_Lucas""context/future extensions.txt" /*_endCmd /********************************************** ; >>>>>>>>> Gaussian versus SI units ; /*$ cat >>"$p_augmented" "$d_Lucas""math nomenclature/Gaussian versus SI units.txt" /*_endCmd /********************** >>>>>> Symbol checking and translation - short description /*$ cat >>"$p_augmented" "$d_Lucas""context/symbols [check, translate].txt" /*_endCmd /******************************************** >>>>>>>>> HFLN = Howells FlatLiner Notation !!!!!!!!!!!!!! 31May2016 /*$ cat >>"$p_augmented" "$d_Lucas""context/Howells flat-line notation short description.txt" /*_endCmd /********************** >>>>>> Document build short description /*$ cat >>"$p_augmented" "$d_Lucas""context/document build short description.txt" /*_endCmd /************************************************* >>>>>> REFERENCES /*$ cat >>"$p_augmented" "$d_Lucas""context/references.txt" /*_endCmd enddoc