www.BillHowell.ca's review of Bill Lucas's math in his book the "Universal Force, Volume I", equationS (4-[32-37])
/*/*$ echo "version= $date_ymdhm, cos - 1 inclusion : $cos_inclusion" >>"$p_augmented"
version= 191025 17h43m, cos - 1 inclusion : yes
This file is :
/*/*$ echo " $p_augmented" >>"$p_augmented"
/media/bill/ramdisk/191025 17h43m math Lucas, 4-32-37, cos - 1 yes.txt
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SUMMARY
This file is a focus on the effects on Lucas's equations (4-32) to (4-37) of one of several different treatments of
∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] :
- included in derivations
- arbitrarily set to zero
/*/*$ cat >>"$p_augmented" "$d_Lucas""context/d-dt Rpcs^-5*t*_cos - 1 comments.txt"
In Lucas's "Universal Force" theory, the expression ∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] has an important influence over the derivation of the "relativistic correction factor" :
{1 - (β*sin(Aθpc(POIo(t),t=0)^n}^(3/2)
A key objective of equations 4-32 through 4-37 is to derive that relativistic correction factor.
Lucas sets ∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] to zero, in which case :
- the correct functional forms of (β*sin(Aθpc(POIo(t),t=0)^n are obtained
- the coefficients of the series [are divergent, are not binomial series]
I personally cannot justify arbitrarily setting ∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] to zero within the context of Lucas's theories. In other words, I have not been able to get the same result as Lucas. That's not a big deal for me, as I suspect that the relativistic correction factor may be more of an [instrumentation, measurement, observation] issue rather than a basic phenomena of electromagnetism.
My attempts up to 16Oct2019 were based on applying an [iterative, non-feedback] approach similar to Newton's method for solving implicit equations. Two variants were attempted :
1. Non-inclusion of expressions with (cos(Aθpc(POIo(t),t=0)) - 1)
- Once I corrected numerous mistakes I had made, these led to the proper functional result, but the coefficients of the infinite series were constantly expanding.
- see sub-directory "$d_Lucas""formulae Lucas/cos - 1 no, iterative, non-feedback/"
2. Non-inclusion of expressions with (cos(Aθpc(POIo(t),t=0)) - 1) - as recommended by Lucas
- The proper functional form was NOT obtained. Again, the coefficients of the infinite series were constantly expanding.
- I do NOT have any solid reason (certainly no proof) that allows the (cos - 1) terms to be dropped! Severely restrictive (sometimes inconcistent) can lead to this, but I am not at all comfortable with these!
- see sub-directory "$d_Lucas""formulae Lucas/cos - 1 yes, iterative, non-feedback/"
In neither case above were the binomial series coefficients obtained.
16Oct2019 Status
That's it. I give up on the relativistic correction factor, from an [iterative, non-feedback] perspective, even though it is quite likely that I have made [simple, fundamental] errors.
Only versions of 4-32 through 4-37 that drop expressions with (cos(Aθpc(POIo(t),t=0)) - 1) are shown below in this document (this could change in the future!). However, you can compare results by looking at :
"$d_Lucas""formulae Lucas/cos - 1 no, iterative, non-feedback/Lucas 4-32-37 no cos - 1.txt"
"$d_Lucas""formulae Lucas/cos - 1 yes, iterative, non-feedback/Lucas 4-32-37 with cos - 1.txt"
But as per the section "Multiple conflicting hypothesis" below, other approaches may also be considered.
I currently think that "∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)]" DEFINITELY should be included in derivations, but that :
- is based on a iterative solution, rather than an electromagnetic feedback effect as per Thomas Barnes (Lucas's source)
- ruins the functional form of results for the relativistic correction factor
- the coefficients of the series [are divergent, are not binomial series]
Note that [special, general] relativity does provide an explanation, but :
- in a non-phenomenological sense (more like data-fitting)
- "General relativity is a turkey" - see ?link to my web-page?
The concepts of [Thomas Barnes, Oleg Jefimenko, Ed Dowdye, Rami Ahmad El-Nabulsi, others] may provide a much more solid explanation for the "relativistic correction factor", but I have not verified these concepts step-by-step as of 24Oct2019.
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To view this file :
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- do NOT use a word processor! That will likely corrupt the files, losing functionality.
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/*------> ∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)]
2015-2016 earlier versions
16Oct2019 clear rational and results for whether or not this term should be included in derivations
/*/*$ cat >>"$p_augmented" "$d_Lucas""math Howell/cos - 1 $cos_inclusion, iterative, non-feedback/d-dt Rpcs^-5*t*_cos - 1.txt"
/home/bill/Lucas - Universal Force/individual formulae developments/d-dt Rpcs^5*t*_cos - 1.txt
www.BillHowell.ca 24Sep2019 initial based on past work
15Oct2019 - WRONG!!!!!! I cannot drop :
∂[∂(t): Rpcs(POIo(t),t=0)^(-5)*(t=0)*(cos(Aθpc(POIo(t),t=0)) - 1)]
= + Rpcs(POIo(t),t=0)^(-5)*(cos(Aθpc(POIo(t),t=0)) - 1)
This file is used for the "correct aqs far as I can tell" derivations of equations (4-[32-37]).
Cool! derivative = original expression without t at t=0 !!!
/********************* 24Sep2019
>>>>>> ∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] = 0
/%
1) ∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)]
a) = ∂[∂(t): Rpcs(POIo(t),t)^(-5)]*t*cos(Aθpc(POIo(t),t))]
b) - ∂[∂(t): Rpcs(POIo(t),t)^(-5) *t]
/*+-----+
/* looking at (1a) /%
2) ∂[∂(t): Rpcs(POIo(t),t)^(-5) *t *cos(Aθpc(POIo(t),t))]
a) = ∂[∂(t): Rpcs(POIo(t),t)^(-5)] *t *cos(Aθpc(POIo(t),t))
b) + Rpcs(POIo(t),t)^(-5) *∂[∂(t): t] *cos(Aθpc(POIo(t),t))
c) + Rpcs(POIo(t),t)^(-5) *t *∂[∂(t): cos(Aθpc(POIo(t),t))]
/* from section "∂[∂(t): Rpcs(POIo(t),t)^(-α)]" : /%
∂[∂(t): Rpcs(POIo(t),t)^(-α)] = α*Vons(PART)*Rpcs(POIo(t),t)^(-α - 1)*cos(Aθpc(POIo(t),t))
/* therefore /%
d) ∂[∂(t): Rpcs(POIo(t),t)^(-5)] = 5*Vons(PART)*Rpcs(POIo(t),t)^(-6)*cos(Aθpc(POIo(t),t))
/* noting /%
e) ∂[∂(t): t] = 1
f) ∂[∂(t): cos(Aθpc(POIo(t),t))] = -sin(Aθpc(POIo(t),t))*∂[∂(t): Aθpc(POIo(t),t))]
/* from "Howell - independent math for Lucas Universal Force, Chapter 4.txt"
section "∂[∂(t): Aθpc(POIo(t),t))]" : /%
g) ∂[∂(t): Aθpc(POIo(t),t)] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)
/* therefore /%
h) ∂[∂(t): cos(Aθpc(POIo(t),t))]
= -sin(Aθpc(POIo(t),t))*Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)
= (-1)*Vons(PART)*sin(Aθpc(POIo(t),t))^2/Rpcs(POIo(t),t)
/* substitute (2d,e,h) into (2a-c) /%
3) ∂[∂(t): Rpcs(POIo(t),t)^(-5) *t *cos(Aθpc(POIo(t),t))]
a) = 5*Vons(PART)*Rpcs(POIo(t),t)^(-6)*cos(Aθpc(POIo(t),t)) *t *cos(Aθpc(POIo(t),t))
b) + Rpcs(POIo(t),t)^(-5) *1 *cos(Aθpc(POIo(t),t))
c) + Rpcs(POIo(t),t)^(-5) *t *(-1)*Vons(PART)*sin(Aθpc(POIo(t),t))^2/Rpcs(POIo(t),t)
/* collecting & rearranging terms /%
4) ∂[∂(t): Rpcs(POIo(t),t)^(-5) *cos(Aθpc(POIo(t),t)) *t ]
a) = 5*Vons(PART)*Rpcs(POIo(t),t)^(-6)*cos(Aθpc(POIo(t),t))^(2) *t
b) + Rpcs(POIo(t),t)^(-5)*cos(Aθpc(POIo(t),t))
c) - Vons(PART)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^(2) *t
/*+-----+
/* looking at (1b) /%
5) ∂[∂(t): Rpcs(POIo(t),t)^(-5) *t]
a) = ∂[∂(t): Rpcs(POIo(t),t)^(-5)] *t
b) + Rpcs(POIo(t),t)^(-5) *∂[∂(t): t]
/* sub (2d,e) into (5a,b) /%
6) ∂[∂(t): Rpcs(POIo(t),t)^(-5) *t]
a) = 5*Vons(PART)*Rpcs(POIo(t),t)^(-6)*cos(Aθpc(POIo(t),t)) *t
b) + Rpcs(POIo(t),t)^(-5) *1
/*+-----+
sub (4,6) into (1)
/%
7) ∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)]
a) = + 5*Vons(PART)*Rpcs(POIo(t),t)^(-6)*cos(Aθpc(POIo(t),t))^(2) *t
b) + Rpcs(POIo(t),t)^(-5)*cos(Aθpc(POIo(t),t))
c) - Vons(PART)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^(2) *t
d) - 5*Vons(PART)*Rpcs(POIo(t),t)^(-6)*cos(Aθpc(POIo(t),t)) *t
e) - Rpcs(POIo(t),t)^(-5)
/* At time t=0 /%
8) ∂[∂(t): Rpcs(POIo(t),t=0)^(-5)*(t=0)*(cos(Aθpc(POIo(t),t=0)) - 1)]
= + Rpcs(POIo(t),t=0)^(-5)* cos(Aθpc(POIo(t),t=0))
- Rpcs(POIo(t),t=0)^(-5)
/* finally /%
(mathH)
∂[∂(t): Rpcs(POIo(t),t=0)^(-5)*(t=0)*(cos(Aθpc(POIo(t),t=0)) - 1)]
= + Rpcs(POIo(t),t=0)^(-5)*(cos(Aθpc(POIo(t),t=0)) - 1)
(endMath)
/*_endCmd
/**********************************************************
/*------> (4-32) EIods(POIo,t=0,1st stage), F therefore E balance - iteration #1 on (4-30)
initial - 15Sep2015, rev1 - ??, rev2 - 25Sep2015
15Sep2015 p71h0.33 Iterate [(4-31) into (4-30)] to obtain solution.
09Jun2016rev5 use HFLN, corrected (I hope) [reference frames, notations, derivatives, integrals]
14Jun2016rev6 fix my error (cosOp - 1) !!
22Aug2019 remove (cosOp - 1) term
30Sep2019 corrections, cleanup
/*/*$ cat >>"$p_augmented" "$d_augment""cos - 1 $cos_inclusion, iterative, non-feedback/04_32 work.txt"
Following the dedication in Lucas's book "Universal Force : Volume 1" :
"... Thomas L. Barnes, professor of Physics at the University of
Texas at El Paso, who showed the way to eliminate Einsteins Special
Relativity Theory from electrodynamics by taking into account the electrical
feedback effects of finite-sized charge particles. ..."
1. using Lucas's (4-30) but "Howells Flat-Liner Notation" (HFLN) - Lucas instructs iterative substitutions
for Ei(ro - vo*t,t) at t=0 implicitly, and dropping v*t*(cosO - 1)=0|t=0 terms as we go.
In this version, I do NOT follow that advice, as I can't see how to justify it.
Notice that the expression variables are ALL scalar
/%
????:(mathL)# 4-30
EIods(POIp(t),t) - EIods(POIp(t),t=0)
= 3*Q*Vons(PART)^2/c^2*Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))*Rpcs(POIo(t),t=0)^( - 5)
*{ Rocs(POIo)*cos(Aθpc(POIo(t),t)) - Vons(PART)*t }
]
+ Vons(PART) /c *Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*∂[∂(t): EIods(POIo,t)]*sin(Aθpc(POIo(t),t)) ]
/* rearranging for ease of view and work :
/%1) EIods(POIp(t),t) - EIods(POIp(t),t=0)
1a) = 3*Q(PART)*Vons^2/c^2*Rocs(POIo)^2*
1a1) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0):
1a1a) { Rocs(POIo)^1*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t=0)^(-5)*sin(Aθpc(POIo(t),t))
1a1b) - Vons(PART)*t *Rpcs(POIo(t),t=0)^(-5)*sin(Aθpc(POIo(t),t))
1a2) }
1b) + Vons(PART) /c *Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*∂[∂(t): EIods(POIo,t)]*sin(Aθpc(POIo(t),t)) ]
/* Substitute for (1a1) through (1a2) from corresponding "Bottom part" results below
/%2) EIods(POIp(t),t) - EIods(POIp(t),t=0)
2a) = 3*Q(PART)*Vons^2/c^2*Rocs(POIo)^2*
2a1)
2a1a) (+ Rocs(POIo) *Rpcs(POIo(t),t=0)^(-5)* sin(Aθpc(POIo(t),t=0))^2/2
2a1b) + Vons(PART)*t*Rpcs(POIo(t),t=0)^(-5)*(cos(Aθpc(POIo(t),t=0)) - 1)
2a2) )
2b) + Vons(PART) /c *Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*∂[∂(t): EIods(POIo,t)]*sin(Aθpc(POIo(t),t)) ]
/* Substitute Vons/c = β, factor out *Rpcs(POIo(t),t=0)^(-5)
/%3) EIods(POIp(t),t) - EIods(POIp(t),t=0)
3a) = 3*Q(PART)*β^2 *Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(-5)*
3a1a) (+ 1/2 *Rocs(POIo) * sin(Aθpc(POIo(t),t=0))^2
3a1b) + Vons(PART)*t *(cos(Aθpc(POIo(t),t=0)) - 1)
3a1c) )
3b) + β^1 *Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*∂[∂(t): EIods(POIo,t)]*sin(Aθpc(POIo(t),t)) ]
/* From (4-31)
/% EIods(POIo,t,Aθpc(POIp) = 0)*Rpch(POIo(t),t) = -λ(Vons(PART))*E0ods(POIo,t)*Rpch(POIo(t),t)
/* Notice that the static component E0 isn't direction dependent, just distance
The above relation also implies that
/%
4) EIods(POIo,t,Aθpc(POIp) = 0) = -λ(Vons(PART))*E0ods(POIo,t)
/* This must be converted to t=0 basis to be compatible, but it is on Aθpc(POIp) basis!!???
No idea of subtleties! Just use t basis that corresponds to Aθpc(POIp) basis?! (not changes with Aθpc(POIp)!?)
/* substitute (4) into (3) above /%
5) EIods(POIo,t=0) + λ(Vons(PART))*E0ods(POIo,t=0)
5a) = 3*Q(PART)*β^2 *Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(-5)
5a1a) *(+ 1/2 *Rocs(POIo) * sin(Aθpc(POIo(t),t=0))^2
5a1b) + Vons(PART)*t *(cos(Aθpc(POIo(t),t=0)) - 1)
5a1c) )
5b) + β*Rocs(POIo)^2*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*∂[∂(t): EIods(POIo,t)]*sin(Aθpc(POIo(t),t)) ]
/* rearranging /%
6) EIods(POIo,t=0,1st stage)
= + 3 *β^2*Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(-5) *1/2*Rocs(POIo)*sin(Aθpc(POIo(t),t=0))^2
+ 3 *β^2*Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(-5) *Vons(PART)*t *(cos(Aθpc(POIo(t),t=0)) - 1)
- λ(Vons(PART))*E0ods(POIo,t=0)
+ β*Rocs(POIo)^2*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))*∂[∂(t): EIods(POIo,t)] ]
/* Compact form : /%
7) EIods(POIo,t=0,1st stage) = K_1st + f_sphereCapSurf(EIods(POIo,t))
/* where : /%
K0 = + 3/2*β^2*Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t=0)^(-5)*sin(Aθpc(POIo(t),t=0))^2
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)
K2 = - λ(Vons(PART))*Q(PART) *Rpcs(POIo(t),t=0)^(-2)
K_1st = K1 + K2
f_sphereCapSurf(x) = β*Rocs(POIo)^2*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))*∂[∂(t): x] ]
/****************************************************************************
>>>>>>>>> Bottom part :
/******+
/* Looking at (1a1) through (1a2) :
/%1a1) ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0):
1a1a) Rocs(POIo)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t=0)^(-5)*sin(Aθpc(POIo(t),t))
1a1b) - Vons(PART)*t *Rpcs(POIo(t),t=0)^(-5)*sin(Aθpc(POIo(t),t))
1a2) ]
/* Distribute the integral
/%1a1)
1a1a) ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): Rocs(POIo)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t=0)^(-5)*sin(Aθpc(POIo(t),t))}
1a1b) ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): - Vons(PART)*t*Rpcs(POIo(t),t=0)^(-5)*sin(Aθpc(POIo(t),t))}
1a2)
/* Integration with respect to "AOtc" from 0 to AOtc(RFt) along curve (not a time derivative!)
see "/media/bill/PROJECTS/Lucas - Universal Force/Howell - Background math for Lucas Universal Force, Chapter 4.txt", "Derivatives & Integrals adapted to Chapter 4"
Remove from integral : [c,β,Q(PART),lambda(v),Vonv,Rocs(POIo),Rpcs(RFt),t,E0ods(POIo,t=0) for integrals (constant during integration!)]
Retain in integral : [terms with AOtc,?E,B,??]
/* Use list of special integrals - see "Howell - Background math for Lucas Universal Force, Chapter 4.odt" for derivation
These are ONLY applicable when integrating from 0 to AOpc, otherwise the lower limit must be addressed!!!
For t=0 RFt applies (RFp & RFo coincide), a,b >=0, integers. POIo does NOT have to be on the perpendicular running through the origins.
For integrals from AOtc=0 to AOtc(POIo,t=0), at AOtc=0, sin(AOpc(RFt))=0 and the expression is 0, producing a zero lower result for definite integrals.
At AOtc(RFt) = 0 & pi/2, the expression is zero. The POIo doesnt have to be at pi/2 to the particle wrt Vonv(PART).
[1 - cos(Aθpc(POIp,t=0))]=0|t=0
for now assume Aθpc = Aθoc
/%1a1)
1a1a) (+ Rocs(POIo) *Rpcs(POIo(t),t=0)^(-5)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)) ]
1a1b) - Vons(PART)*t*Rpcs(POIo(t),t=0)^(-5)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))}
1a2) )
/* first definite integral : ∫(dOp, 0 to Opf : sinOp*cosOp ) = sin^2Op/2
/% ∫[∂(Aθpc),0 to Aθpc(POIo(t),t): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)) ]
= sin(Aθpc(POIo(t),t))^2/2 from Aθpc = 0 to Aθoc(POIp(t),t=0)
= sin(Aθpc(POIo(t),t=0))^2/2 - sin(0)
= sin(Aθpc(POIo(t),t=0))^2/2
/* second definite integral : 01Oct2019 - is this a NEGATIVE cos? - YES dcos = -sin, dsin = cos
/% ∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))}
= (-1)*cos(Aθpc(POIo(t),t)) from Aθpc = 0 to Aθoc(POIp(t),t=0)
= (-1)*{ cos(Aθpc(POIo(t),t=0)) - cos(0) }
= (-1)*{ cos(Aθpc(POIo(t),t=0)) - 1 }
/* The full expression becomes /%
( + Rocs(POIo) *Rpcs(POIo(t),t=0)^(-5)* sin(Aθpc(POIo(t),t=0))^2/2
- Vons(PART)*t*Rpcs(POIo(t),t=0)^(-5)*(-1)*{ cos(Aθpc(POIo(t),t=0)) - 1 }
)
/* or /%
1a1a) (+ Rocs(POIo) *Rpcs(POIo(t),t=0)^(-5)* sin(Aθpc(POIo(t),t=0))^2/2
1a1b) + Vons(PART)*t*Rpcs(POIo(t),t=0)^(-5)*(cos(Aθpc(POIo(t),t=0)) - 1)
1a2) )
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Likely_Lucas_error_or_omission
04_32
F therefore E balance - simplified (4-30)
/$L Eis(r - v*t,t)|t=0 + λ(v)*E0s(r - v*t,t)|t=0
= 3 *β*rs *q/|r - v*t|^5*{rs/2*sin(θ´)^2 + vs*t*(cos(θ´) - 1)}|t=0
+ β*rs ^2 *∫[∂(θ´),0 to θ´f: 1/rs/c*∂[∂(t): Eis(r - v*t,t)]*sin(θ´))|t=0
/%L Eis(r - v*t,t)|t=0 + λ(v)*E0s(r - v*t,t)|t=0
= 3*(β*rs)^2*q/|r - v*t|^5*{rs/2*sin(θ´)^2 + vs*t*(cos(θ´) - 1)}|t=0
+ β*rs ^2 *∫[∂(θ´),0 to θ´f: 1/rs/c*∂[∂(t): Eis(r - v*t,t)]*sin(θ´))|t=0
/* Lucas derivation - Compact form : /%
EIods(POIo,t=0) = K0 + K1 + K2 + f_sphereCapSurf(EIods(POIo,t))
/* Howell derivation - drop the K1 as it fall out and doesn't contribute at t=0, see 4-33 /%
(mathL) EIods(POIo,t=0,1st stage) = K_1st + f_sphereCapSurf(EIods(POIo,t))
(endMath)
(mathL)/* 4-32 /%
EIods(POIo,t=0,1st stage)
= + 3 *β^2*Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(-5) *1/2*Rocs(POIo)*sin(Aθpc(POIo(t),t=0))^2
+ 3 *β^2*Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(-5) *Vons(PART)*t *(cos(Aθpc(POIo(t),t=0)) - 1)
- λ(Vons(PART))*E0ods(POIo,t=0)
+ β*Rocs(POIo)^2*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))*∂[∂(t): EIods(POIo,t)] ]
(endMath)
/* WRONG??
I have (β*rs)^2 rather than β*rs in first expression RHS (?), but in Lucas (4-33) the r^2 seems to "reappear" !?!?
/*_file_insert_path "$d_Lucas""relativistic factor, restrictive conditions.txt"
# enddoc
/*_endCmd
/**********************************************************
/*------> (4-33) EIods(POIo,t=0, 2nd stage), F therefore E balance
29May2016 I had mistakenly used Ei in place of Eis, so this has been corrected in the final result.
14Jun2016rev3 fix my error (cosOp - 1) !!
Lucas"s version of (4-32)
(4-32rev3 F therefore E balance - simplified (4-30)
22Aug2019 start revision used revamped (4-32), finished 26Aug2019
03Sep2019 fix error with Equation (5) below - dropped a term, K3 was incorrect then simply dropped from expressions
24Sep2019 drop K1, add "Highly restrictive conditions"
30Sep2019 correct + sign for K1
/*/*$ cat >>"$p_augmented" "$d_augment""cos - 1 $cos_inclusion, iterative, non-feedback/04_33 work.txt"
/* USUALLY, I WORK FROM LUCAS'S RESULTS RATHER THAN MY OWN, BUT IN THIS CASE I WILL START WITH MINE
22Aug2019 Howell's version of (4-32) :
/%
(mathL)/* generative form /%
EIods(POIo,t,2nd stage) = K_1st + f_sphereCapSurf(EIods(POIo,t,1st stage))
(endMath)
/* where : /%
K_1st = + 3/2*β^2*Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t=0)^(-5)*sin(Aθpc(POIo(t),t=0))^2
- λ(Vons(PART)) *Q(PART) *Rpcs(POIo(t),t=0)^(-2)
f_sphereCapSurf{x} = β*Rocs(POIo)^2*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))*∂[∂(t): x] ]
/%
/* 1. So what is the next step?
It is interesting to directly compare (4-32c), as labelled (a), with (4-30).
(4-32) does not have the same integral that was replaced in (4-30), so the next target appears to be the integral term in the 2nd expression on the RHS :
using /%
(1)
????:(mathL)# 4-32
EIods(POIo,t=0,1st stage)
= + 3 *β^2*Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(-5) *1/2*Rocs(POIo)*sin(Aθpc(POIo(t),t=0))^2
+ 3 *β^2*Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(-5) *Vons(PART)*t *(cos(Aθpc(POIo(t),t=0)) - 1)
- λ(Vons(PART))*E0ods(POIo,t=0)
+ β*Rocs(POIo)^2*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))*∂[∂(t): EIods(POIo,t)] ]
/* Take the derivative of (1) /%
3) ∂[∂(t): EIods(POIo,t,1st stage)]
= + ∂[∂(t): 3/2*β^2 *Q(PART)*Rocs(POIo)^3 *Rpcs(POIo(t),t=0)^(-5) * sin(Aθpc(POIo(t),t=0))^2 ]
+ ∂[∂(t): 3 *β^2 *Q(PART)*Rocs(POIo)^2 *Rpcs(POIo(t),t=0)^(-5)*Vons(PART)*t*(cos(Aθpc(POIo(t),t=0)) - 1) ]
- ∂[∂(t): λ(Vons(PART)) *Q(PART) *Rpcs(POIo(t),t=0)^(-2) ]
+ ∂[∂(t): f_sphereCapSurf(EIods(POIo,t,0th stage)) ]
/* Putting (3) back into (1) Yields : /%
4) EIods(POIo,t=0,2nd stage)
= + K_1st
+ β^1*Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
*{
+ ∂[∂(t): 3/2*β^2 *Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t=0)^(-5) * sin(Aθpc(POIo(t),t=0))^2 ]
+ ∂[∂(t): 3 *β^2 *Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(-5)*Vons(PART)*t*(cos(Aθpc(POIo(t),t=0)) - 1) ]
- ∂[∂(t): λ(Vons(PART)) *Q(PART) *Rpcs(POIo(t),t=0)^(-2) ]
+ ∂[∂(t): f_sphereCapSurf(EIods(POIo,t,0th stage)) ]
}
]
5) EIods(POIo,t=0,2nd stage)
= + K_1st
+ β^1*Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
*{
+ ∂[∂(t): 3/2*β^2 *Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t=0)^(-5) *sin(Aθpc(POIo(t),t=0))^2 ]
+ ∂[∂(t): 3 *β^2 *Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(-5)*Vons(PART)*t*(cos(Aθpc(POIo(t),t=0)) - 1) ]
- ∂[∂(t): λ(Vons(PART)) *Q(PART) *Rpcs(POIo(t),t=0)^(-2) ]
}
]
+ f_sphereCapSurf{∂[∂(t): f_sphereCapSurf(EIods(POIo,t,0th stage))]}
/* for symbols & HIGHLY restrictive conditions :
see "textIn - relativistic factor, intermediate symbols, restrictive conditions.txt"
/%
/* |--???--> (4-33) I DON'T GET THIS! : E0ods(POIo,t=0) = Q(PART)*Rocs(POIo)/Rpcs(POIo(t),t=0)^3
It appears that, from (4-8) Bi(r,v,t), Lucas has replaced E0(ro - vo*t,t) in the last term with
The Grassman form of the generalized Ampere force law is based on derivations in Appendix A (eq (A19).
(4-08) is the derivation of (4-01) from the Grassman/Biot-Savart form of Amperes Law
This is derived in Appendix A...
/$ q/c*(vr´)/rs'^3 = (v/c)E0(r',t')
/* reference : Jackson 1999 p?? Eqn ?? (I lost the reference location, cant find!!
such that (in Gaussian coordinates?)
This does NOT follow! :
/$ E0(r,t) = q*r´/r´s^3 = q*r´/|r - v*t|^3
/* BUT - in (4-33), Lucas has r rather than r' in numerator, WHICH SEEMS WRONG :
/$ E0(r,t) = q*r /r´s^3 = q*r /|r - v*t|^3
/* translate reference frame :
/% E0ods(POIo,t=0)
= Q(PART)*Rocs(POIo)/Rpcs(POIo(t),t=0)^3
/* <--???--|
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn question with respect to E0ods(POIo,t=0) expression!
04_33 22Aug2019 start revision, 27Aug2019 finished revision
F therefore E balance - iterations on (4-32)
/$L Eis(r - v*t,t) APPLY |t=0 TO EACH TERM
= K0 + K2
+ β*rs^2*∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t): K0 + K2])
+ β*rs^2*∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t): K3])
/* where :
/$L K0 + K2 = 3/2*β^2*q*rs^3/|r - v*t|^5*sin(θ´)^2 - λ(v)*q*rs/|r - v*t|^3
K3 = β *rs^2*∫[∂(θ´),0 to Of: 1/rs/c*sin(θ´)*∂[∂(t): Eis(r - v*t,t)])
/* 03Sep2019 This is old!
/$H Eis(r - v*t,t) APPLY |t=0 TO EACH TERM
= K0 + K2
+ β*rs^2*∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t): K0 + K2])
+ β*rs^2*∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t): K3])
/* where :
/$H K0 + K2 = 3/2*β^2*q*rs^3/|r - v*t|^5*sin(θ´)^2 - λ(v)*q/|r - v*t|^2
K3 = β *rs^2*∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ)*∂[∂(t): Eis(r - v*t,t)])
/* OK - works great by using a blend of Lucas & Howell expressions for (4-32). This assumes a Lucas typo in 4-30, dropping a power of r
EXPLAIN :
Lucas states p71h0.25 that the v*t*(cosO - 1) are dropped, Presumably, at t=0 cosθ = 1, so (cosO - 1)|t=0 = 0.
/* Result 14Sep2019 - Compact form /%
(mathL)/* 4-33 /%
EIods(POIo,t=0,2nd stage)
= + K_1st
+ β^1*Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
*{
+ ∂[∂(t): 3/2*β^2 *Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t=0)^(-5) *sin(Aθpc(POIo(t),t=0))^2 ]
+ ∂[∂(t): 3 *β^2 *Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(-5)*Vons(PART)*t*(cos(Aθpc(POIo(t),t=0)) - 1) ]
- ∂[∂(t): λ(Vons(PART)) *Q(PART) *Rpcs(POIo(t),t=0)^(-2) ]
}
]
+ f_sphereCapSurf{∂[∂(t): f_sphereCapSurf(EIods(POIo,t,0th stage))]}
(endMath)
/* for symbols & HIGHLY restrictive conditions :
see "textIn - relativistic factor, intermediate symbols, restrictive conditions.txt"
where : /%
K_1st = + 3/2*β^2*Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t=0)^(-5)*sin(Aθpc(POIo(t),t=0))^2
- λ(Vons(PART)) *Q(PART) *Rpcs(POIo(t),t=0)^(-2)
f_sphereCapSurf{x} = β*Rocs(POIo)^2*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))*∂[∂(t): x] ]
/%
/* Note that this is a clean definition of a non-recursive (forward) process :
EIods(POIo,t=0, zeroth stage) = + K0 + K2 + f_sphereCapSurf(EIods(POIo,t))
EIods(POIo,t=0, ith stage) = + K0 + K2 + f_sphereCapSurf(K0 + K2 + f_sphereCapSurf(EIods(POIo,t=0,(i-1) stage))) )
# enddoc
/*_endCmd
/**********************************************************
/*------> (4-34) EIods(POIo,t=0, 2nd stage), K_2nd from taking partial derivatives wrt time
for previous versions, see "Howell - Old math of Lucas Universal Force.ndf"
16Sep2015 1st or 2nd?, 20Sep2015 rev2, 25Sep2015 rev4
29May2016rev5
14May2016rev6
13Aug2019 !!! ∂[∂(x): sin(x)] = cos(x) !!! ????????????????????????? /%
27Aug2019 revamp, hopefully fixing wrong [+,-] issues for (4-37)
There is an ambiguity in my expressions with [t, t=0] - sometimes I've left in the wrong form
03Sep2019 fix error in K2 term from 04_33 (actually - this was not a problem?)
03Sep2019 Note that I corrected the results there, which had Aθpc(POIp(t),t), which is an illegal symbol!!!
this was corrected to Aθoc(POIp(t),t) or Aθpc(POIo(t),t)
03Sep2019 no net change to end result of 04_34
24Sep2019 following changes to 4-33, drop K1, add "Highly restrictive conditions"
02Oct2019 fix ∂[∂(t): Rpcs(POIo(t),t)^(-b)*sin(Aθpc(POIo(t),t))^a], add K_1st term
14Oct2019 fix generative form for 4-34 (forgot to do that earlier)
/*/*$ cat >>"$p_augmented" "$d_augment""cos - 1 $cos_inclusion, iterative, non-feedback/04_34 work.txt"
/* Restating iterative solution :
(mathL)/* generative form /%
EIods(POIo,t,2nd stage) = K_1st + f_sphereCapSurf(EIods(POIo,t,1st stage))
(endMath)
/* using /%
1)
????:(mathL)
EIods(POIo,t=0,2nd stage)
= + K_1st
+ β^1*Rocs(POIo)^2
1a) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
*{
+ ∂[∂(t): 3/2*β^2 *Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t)^(-5) *sin(Aθpc(POIo(t),t))^2 ]
+ ∂[∂(t): 3 *β^2 *Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t)^(-5)*Vons(PART)*t*(cos(Aθpc(POIo(t),t)) - 1) ]
- ∂[∂(t): λ(Vons(PART)) *Q(PART) *Rpcs(POIo(t),t)^(-2) ]
}
]
+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}
??????????????????????????????????
/* Substitute for (1b) from "Bottom-up (6b)" results below
2) EIods(POIo,t=0,2nd stage)
= + K_1st
+ β*Rocs(POIo)^2
1g) *{ + 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)) ]
+ 3 *β^3*Q(PART)*Rocs(POIo) *Rpcs(POIo(t),t)^(-5) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t)) *(cos(Aθpc(POIo(t),t)) - 1)
- 2 *λ(Vons(PART)) *β *Q(PART)/Rocs(POIo) *Rpcs(POIo(t),t)^(-3) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t)) *cos(Aθpc(POIo(t),t)) ]
}
+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}
/***********************************
>>>>>> Bottom-up (2b1)
/* Looking at (1b) /%
1b) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
*{
+ ∂[∂(t): 3/2*β^2 *Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t)^(-5) *sin(Aθpc(POIo(t),t=0))^2 ]
+ ∂[∂(t): 3 *β^2 *Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t)^(-5)*Vons(PART)*t*(cos(Aθpc(POIo(t),t)) - 1) ]
- ∂[∂(t): λ(Vons(PART)) *Q(PART) *Rpcs(POIo(t),t)^(-2) ]
}
]
/* "Percolate" constant terms up through [derivatives, integrals] in Chapter 4 :
see "Howell - Background math for Lucas Universal Force, Chapter 4.txt"
section "Constants of [derivative, integration] expressions"
section "Rpcs(POIo(t),t)" is a constant for integrals like "∫{∂(Aθtc),0 to Aθoc(POIp(t),t=0):" ?"
[c, β, λ(Vons(PART)), Q(PART), Rocs(POIo), Vons(PART)]
[Rpcs(POIo(t),t)]
- is NOT a constant wrt ∂[∂(t):
- IS a constant wrt ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0):
/%
1c) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
*{ + 3/2 *β^2 *Q(PART)*Rocs(POIo)^3*∂[∂(t): Rpcs(POIo(t),t)^(-5) *sin(Aθpc(POIo(t),t))^2 ]
+ 3 *β^2*Vons(PART)*Q(PART)*Rocs(POIo)^2*∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t))] - 1) ]
- λ(Vons(PART)) *Q(PART) *∂[∂(t): Rpcs(POIo(t),t)^(-2) ]
}
]
/* using /%
2304:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-5)] = 7*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6)
????:(mathH)
∂[∂(t): Rpcs(POIo(t),t=0)^(-5)*(t=0)*(cos(Aθpc(POIo(t),t)) - 1)]
= + Rpcs(POIo(t),t=0)^(-5)*(cos(Aθpc(POIo(t),t)) - 1)
2059:(mathL) ∂[∂(t): Rpcs(POIo(t),t)^(-2)] = 2*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3)
/* substitute into (1b) /%
1d) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
*{ + 3/2 *β^2 *Q(PART)*Rocs(POIo)^3 *7*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6)
+ 3 *β^2*Vons(PART)*Q(PART)*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-5)*(cos(Aθpc(POIo(t),t)) - 1)
- λ(Vons(PART)) *Q(PART) *2*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3)
}
]
/* [collect, rearrange] terms /%
1e) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
*{ + 3/2*7*β^2 *Vons(PART)*Q(PART)*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))
+ 3 *β^2 *Vons(PART)*Q(PART)*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-5)*(cos(Aθpc(POIo(t),t)) - 1)
- 2*λ(Vons(PART)) *Vons(PART)*Q(PART) *Rpcs(POIo(t),t)^(-3)*cos(Aθpc(POIo(t),t))
}
]
/* distribute the integral
/* "Percolate" constant terms up through [derivatives, integrals] in Chapter 4 :
see "Howell - Background math for Lucas Universal Force, Chapter 4.txt"
section "Constants of [derivative, integration] expressions"
section "Rpcs(POIo(t),t)" is a constant for integrals like "∫{∂(Aθtc),0 to Aθoc(POIp(t),t=0):" ?"
[c, β, λ(Vons(PART)), Q(PART), Rocs(POIo), Vons(PART)]
[Rpcs(POIo(t),t)]
- is NOT a constant wrt ∂[∂(t):
- IS a constant wrt ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0):
/%
1f) *{ + 21/2*β^2*Rocs(POIo)^3 *Q(PART)*Vons(PART) *1/c/Rocs(POIo)*Rpcs(POIo(t),t)^(-6) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^2 ]
+ 3 *β^2*Vons(PART) *Q(PART)*Rocs(POIo)^2 *1/c/Rocs(POIo)*Rpcs(POIo(t),t)^(-5) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))*(cos(Aθpc(POIo(t),t)) - 1)
- 2 *λ(Vons(PART)) *Q(PART)*Vons(PART) *1/c/Rocs(POIo)*Rpcs(POIo(t),t)^(-3) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t)) ]
}
/* set Vons(PART)/c = β, collect terms /%
1g) *{ + 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)) ]
+ 3 *β^3*Q(PART)*Rocs(POIo) *Rpcs(POIo(t),t)^(-5) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t)) *(cos(Aθpc(POIo(t),t)) - 1)
- 2 *λ(Vons(PART)) *β *Q(PART)/Rocs(POIo) *Rpcs(POIo(t),t)^(-3) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t)) *cos(Aθpc(POIo(t),t)) ]
}
/* insert this result in "Top-down" section above
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Likely_Lucas_error_or_omission
04_34
F therefore E balance - taking partial derivatives wrt time
/$L Eis(r - v*t,t) APPLY |t=0 TO EACH TERM
=+ 3/2*β^2*q*rs^3/|r - v*t|^5*sin(θ´)^2 - λ(v)*q*rs/|r - v*t|^3
+ β *rs^2*∫[∂(θ´),0 to θ´f: sin(θ´)*
(+ 15/2*β^3*q*rs^4/|r - v*t|^7*sin(θ´)^2*cos(θ´)
- 3 *β *q*rs^2/|r - v*t|^5*λ(v) *cos(θ´)
))
+ β^2 *rs^4*∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t):
∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t): Eis(r - v*t,t)])])
/$H Eis(r - v*t,t) APPLY |t=0 TO EACH TERM
=+ 3/2*β^2*q*rs^3/|r - v*t|^5*sin(θ´)^2 - λ(v)*q*rs/|r - v*t|^3
+ β *rs^2*∫[∂(θ´),0 to θ´f: sin(θ´)*
(+ 15/2*β^3*q*rs^3/|r - v*t|^7*sin(θ´)^2*cos(θ´)
- 3 *β *q*rs^1/|r - v*t|^5*λ(v) *cos(θ´)
))
+ β^2 *rs^4*∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t):
∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t): Eis(r - v*t,t)])])
/%H
(mathL)/* 4-34 /%
EIods(POIo,t=0,2nd stage)
= + K_1st
+ β*Rocs(POIo)^2
*{ + 21/2 *β^3*Q(PART)*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-6) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^3 *cos(Aθpc(POIo(t),t)) ]
+ 3 *β^3*Q(PART)*Rocs(POIo) *Rpcs(POIo(t),t)^(-5) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t)) *(cos(Aθpc(POIo(t),t)) - 1)
- 2 *λ(Vons(PART)) *β *Q(PART)/Rocs(POIo) *Rpcs(POIo(t),t)^(-3) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t)) *cos(Aθpc(POIo(t),t)) ]
}
+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}
(endMath)
/* iterative form, see "Howell - independent math for Lucas Universal Force, Chapter 4.txt" /%
1049:(mathH) EIods(POIo,t=0,ith stage) = K_1st + f_sphereCapSurf(EIods(POIo,t=0,(i-1) stage)))}
1046:(mathH) EIods(POIo,t=0,2nd stage) = K_1st + f_sphereCapSurf(EIods(POIo,t=0,1st stage)
/* for symbols & HIGHLY restrictive conditions :
see "textIn - relativistic factor, intermediate symbols, restrictive conditions.txt"
/%
/* for a detailed comparison, see above "Comparison of Lucas (4-34) and previous (4-34) derivation to (5)"
/*_endCmd
/**********************************************************
/*------> (4-35) E0ods(POIo,t) truncated expression with ONLY E0ods(POIo,t) terms
16Sep2015 1st version, ?17Sep2015? rev1, 21Sep2015 rev2, 26Sep2015 rev3
for previous versions, see "Howell - Old math of Lucas Universal Force.ndf"
??? 04Jan2016 Check if should be redone with "Howell - Key math info & derivations for Lucas Universal Force.odt"
29Aug2019 re-derive based on recent work from (4-32) to (4-34)
03Sep2019 fix ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))*cos(Aθpc(POIp(t),t)) ] :
I had sin(Aθpc(POIo(t),t=0))^1/1, should have been sin(Aθpc(POIo(t),t=0))^2/2
24Sep2019 following changes to 4-33, drop K1, add "Highly restrictive conditions"
02Oct2019 percolate fix to ∂[∂(t): Rpcs(POIo(t),t)^(-b)*sin(Aθpc(POIo(t),t))^a], note K_2nd term
/*/*$ cat >>"$p_augmented" "$d_augment""cos - 1 $cos_inclusion, iterative, non-feedback/04_35 work.txt"
/* Objective - expression for : /%
1) ETods(POIo,t) = E0ods(POIo,t) + EIods(POIo,t)
/* Gaussian coordinates - see also 04_33 Equation (7), see [explanation, worry] below /%
2) E0ods(POIo,t) = Q(PART)*Rpcs(POIp)^(-2)
3)
????:(mathL)# 4-34
EIods(POIo,t=0,2nd stage)
= + K_1st
+ β*Rocs(POIo)^2
*{ + 21/2 *β^3*Q(PART)*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-6) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^3 *cos(Aθpc(POIo(t),t)) ]
+ 3 *β^3*Q(PART)*Rocs(POIo) *Rpcs(POIo(t),t)^(-5) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t)) *(cos(Aθpc(POIo(t),t)) - 1)
- 2 *λ(Vons(PART)) *β *Q(PART)/Rocs(POIo) *Rpcs(POIo(t),t)^(-3) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t)) *cos(Aθpc(POIo(t),t)) ]
}
+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}
= + K_1st
+ β*Rocs(POIo)^2
*{ + 21/2 *β^3*Q(PART)*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-6) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^3 *cos(Aθpc(POIo(t),t)) ]
+ 3 *β^3*Q(PART)*Rocs(POIo) *Rpcs(POIo(t),t)^(-5)*{ + ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t)) *cos(Aθpc(POIo(t),t)) ]
- ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t)) ]
}
- 2 *λ(Vons(PART)) *β *Q(PART)/Rocs(POIo) *Rpcs(POIo(t),t)^(-3) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t)) *cos(Aθpc(POIo(t),t)) ]
}
+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}
/* using /%
2687:(mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^3] = sin(Aθpc(POIo(t),t=0))^4/4
3211:(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
????:(mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))] = {1 - cos(Aθpc(POIo(t),t=0)}
2685:(mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^1] = sin(Aθpc(POIo(t),t=0))^2/2
4) EIods(POIo,t=0,2nd stage)
= + K_1st
+ β*Rocs(POIo)^2
*{ + 21/2 *β^3*Q(PART)*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-6) *sin(Aθpc(POIo(t),t=0))^4/4
+ 3 *β^3*Q(PART)*Rocs(POIo) *Rpcs(POIo(t),t)^(-5)*{ + sin(Aθpc(POIo(t),t=0))^2/2
- {1 - cos(Aθpc(POIo(t),t=0)}
}
- 2 *λ(Vons(PART)) *β *Q(PART)/Rocs(POIo) *Rpcs(POIo(t),t)^(-3) *sin(Aθpc(POIo(t),t=0))^2/2
}
+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}
5) EIods(POIo,t=0,2nd stage)
= + K_1st
+ β*Rocs(POIo)^2 *21/2 *β^3*Q(PART)*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4/4
+ β*Rocs(POIo)^2 *3 *β^3*Q(PART)*Rocs(POIo) *Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t=0))^2/2
+ β*Rocs(POIo)^2 *3 *β^3*Q(PART)*Rocs(POIo) *Rpcs(POIo(t),t)^(-5)*{cos(Aθpc(POIo(t),t=0) - 1}
- β*Rocs(POIo)^2 *2 *β *Q(PART)/Rocs(POIo) *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t=0))^2/2 *λ(Vons(PART))
+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}
/* collect factors, rearrange using /%
1108:(mathH)/* differentiable form /%
K_1st
= + Q(PART)
*( 3/2 *β^2*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2
- λ(Vons(PART)) *Rpcs(POIo(t),t)^(-2)
)
6) EIods(POIo,t=0,2nd stage)
= + Q(PART)
*( 3/2 *β^2*Rocs(POIo)^3 *sin(Aθpc(POIo(t),t=0))^2 *Rpcs(POIo(t),t=0)^(-5)
+ 21/8 *β^4*Rocs(POIo)^4 *sin(Aθpc(POIo(t),t=0))^4 *Rpcs(POIo(t),t=0)^(-6)
+ 3 *β^4*Rocs(POIo)^3 *{cos(Aθpc(POIo(t),t=0) - 1} *Rpcs(POIo(t),t=0)^(-5)
- λ(Vons(PART)) *1 *Rpcs(POIo(t),t=0)^(-2)
- λ(Vons(PART)) *1 *β^2*Rocs(POIo) *sin(Aθpc(POIo(t),t=0))^2 *Rpcs(POIo(t),t=0)^(-3)
)
+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}
/* what Lucas seems to have done here is to use : /%
1205:(mathH) E0ods(POIo,t) = E0pds(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2
(mathL)/* HIGHLY restrictive conditions : t=0, RFo(t=0) = Rfp(t=0), Rocs(POIo) = Rpcs(POIo(t),t=0)
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"
/%
Rocs(POIo) = Rpcs(POIo(t),t=0)
(endMath)
/* |--???-->
02Sep2019 Note that I find that the assumption {Rocs(POIo) = Rpcs(POIo(t),t=0)} is a HUGE restriction,
and means to me that the result is not at all general (but neither are standard theories).
<--???--|
/* re-express (6) using Rocs(POIo) = Rpcs(POIo(t),t=0) /%
7) EIods(POIo,t=0,2nd stage)
= + 3/2 *β^2*Q(PART)*Rpcs(POIo(t),t=0)^(-2) *sin(Aθpc(POIo(t),t=0))^2
+ 21/8 *β^4*Q(PART)*Rpcs(POIo(t),t=0)^(-2) *sin(Aθpc(POIo(t),t=0))^4
+ 3 *β^4*Q(PART)*Rpcs(POIo(t),t=0)^(-2)*{cos(Aθpc(POIo(t),t=0) - 1}
- λ(Vons(PART)) *1 *Q(PART)*Rpcs(POIo(t),t=0)^(-2)
- λ(Vons(PART)) *1 *β^2*Q(PART)*Rpcs(POIo(t),t=0)^(-2) *sin(Aθpc(POIo(t),t=0))^2
+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}
/* rearranging /%
8) EIods(POIo,t=0,2nd stage)
= + Q(PART)*Rpcs(POIo(t),t=0)^(-2)
*( + 3/2 *β^2*sin(Aθpc(POIo(t),t=0))^2
+ 21/8 *β^4*sin(Aθpc(POIo(t),t=0))^4
+ 3 *β^4*{cos(Aθpc(POIo(t),t=0) - 1}
- λ(Vons(PART)) *1
- λ(Vons(PART)) *1 *β^2*sin(Aθpc(POIo(t),t=0))^2
)
+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}
= + Q(PART)*Rpcs(POIo(t),t=0)^(-2)
*( + 3/2 *β^2*sin(Aθpc(POIo(t),t=0))^2
+ 21/8 *β^4*sin(Aθpc(POIo(t),t=0))^4
+ 3 *β^4*cos(Aθpc(POIo(t),t=0)
- 3 *β^4
- λ(Vons(PART)) *1
- λ(Vons(PART)) *1 *β^2*sin(Aθpc(POIo(t),t=0))^2
)
+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}
/* now using /%
1174:(mathH) E0pds(POIp) = Q(PART)/Rpcs(POIp)^2
/* substitute (2) into (8), and (8) into (1), drop f_sphereCapSurf expression /%
9) EIods(POIo,t=0,2nd stage)
= + E0ods(POIo,t) *3/2 *β^2*sin(Aθpc(POIo(t),t=0))^2
+ E0ods(POIo,t) *21/8 *β^4*sin(Aθpc(POIo(t),t=0))^4
+ E0ods(POIo,t) *3 *β^4*cos(Aθpc(POIo(t),t=0)
- E0ods(POIo,t) *3 *β^4
- E0ods(POIo,t) *λ(Vons(PART)) *1
- E0ods(POIo,t) *λ(Vons(PART)) *1 *β^2*sin(Aθpc(POIo(t),t=0))^2
/* re-express (9) /%
10) EIods(POIo,t=0,2nd stage)
= + E0ods(POIo,t) *{ 3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 + 21/8*β^4*sin(Aθpc(POIo(t),t=0))^4 + 3*β^4*cos(Aθpc(POIo(t),t=0) - 3*β^4}
- E0ods(POIo,t)*λ(Vons(PART)) *{ 1 + β^2*sin(Aθpc(POIo(t),t=0))^2 }
/* For total E, ETods(POIo,t) /%
11) ETods(POIo,t=0,2nd stage)
= E0ods(POIo,t) + EIods(POIo,t)
= + E0ods(POIo,t) *{ 1 + 3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 + 21/8*β^4*sin(Aθpc(POIo(t),t=0))^4 + 3*β^4*cos(Aθpc(POIo(t),t=0) - 3*β^4}
- E0ods(POIo,t)*λ(Vons(PART)) *{ 1 + β^2*sin(Aθpc(POIo(t),t=0))^2 }
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn 03Sep2019 NOT the same thing!!!
04_35
F therefore E balance - more iterations
all measures at t=0
/$L Ei(r)|t=0
= + E0(rs) *( 3/2*β^2*sin(θ)^2 + 15/8*β^4*sin(θ)^4 )
- E0(rs)*λ(v)*(1 + 3/2*β^2*sin(θ)^2)
/$H Ei(rs,vs) all measures at t=0
= + E0(rs) *( 3/2*β^2*sin(θ´)^2 + 15/8*β^4*sin(θ´)^4 )
- E0(rs)*λ(v)*(1 + 3/2*β^2*sin(θ´)^2)
/* OK other than Lucas EI/ET mixup, Oo/Op limit used by Lucas p72h0.0? probably OK, check later
E0 is added to get ET in 4-36.
As part of the iterative approach, Lucas dropped the last expression with Ei(r).
This is in need of [explanation, clarification] - i.e. this is the first-step approximation only.
/* Both [induced, total] forms are shown below
/%H
/* remove "=0" from t to get :
(mathL)/* differentiable form EIods(POIo,t,2nd stage) /%
EIods(POIo,t,2nd stage)
= + Q(PART)
*( + 3/2 *β^2*sin(Aθpc(POIo(t),t=0))^2 *Rpcs(POIo(t),t=0)^(-2)
+ 21/8 *β^4*sin(Aθpc(POIo(t),t=0))^4 *Rpcs(POIo(t),t=0)^(-2)
+ 3 *β^4*cos(Aθpc(POIo(t),t=0) *Rpcs(POIo(t),t=0)^(-2)
- 3 *β^4 *Rpcs(POIo(t),t=0)^(-2)
- λ(Vons(PART)) *1 *Rpcs(POIo(t),t=0)^(-2)
- λ(Vons(PART)) *1 *β^2*sin(Aθpc(POIo(t),t=0))^2 *Rpcs(POIo(t),t=0)^(-2)
)
+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}
(endMath)
(mathL)
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 + 3*β^4*cos(Aθpc(POIo(t),t=0) - 3*β^4}
- E0ods(POIo,t)*λ(Vons(PART)) *{ 1 + β^2*sin(Aθpc(POIo(t),t=0))^2 }
(endMath)
(mathL)
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 + 3*β^4*cos(Aθpc(POIo(t),t=0) - 3*β^4}
- E0ods(POIo,t)*λ(Vons(PART)) *{ 1 + β^2*sin(Aθpc(POIo(t),t=0))^2 }
(endMath)
/* 16Oct2019 4-35 There is no sense going any further, as I've done similar derivations many times befoire, and this isn't working [functionally, coefficients]!!!
Note that Lucas (4-35) is actually for EIods(POIo,t=0), not ETods(POIo,t).
/* 03Sep2019 the numerical factors differ from Lucas - hopefully that will self-fix with further iterations?
# enddoc
/*_endCmd
/**********************************************************
/*------> (4-36) ETods(POIo,t) expression with ONLY ETods(POIo,t) terms
17Sep2015 2nd iteration
03Sep2019 Redone with HFLN - it now works!!!!!
/*/*$ cat >>"$p_augmented" "$d_augment""cos - 1 $cos_inclusion, iterative, non-feedback/04_36 work.txt"
Note that Lucas (4-35) is actually for EIods(POIo,t=0), not ETods(POIo,t).
I did both in (4-35), so just refer to it for the derivations.
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn No sense doing this, it fails!!
04_36
Er in terms of E0(rs) and L(vs)
/$L ET(rs,vs) = E0(rs) + Ei(rs,vs) all measures at t=0
= E0(rs) *(1 + 3/2*β^2*sin(θ)^2 + 15/8*β^4*sin(θ)^4 )
- E0(rs)*L(vs)*(1 + 3/2*β^2*sin(θ)^2 )
/$H ET(rs,vs) = E0(rs) + Ei(rs,vs) all measures at t=0
= + E0(rs) *(1 + 3/2*β^2*sin(θ)^2 + 15/8*β^4*sin(θ)^4 )
- E0(rs)*L(vs)*(1 + 3/2*β^2*sin(θ)^2)
/* OK - very simple. /%
ETods(POIo,t=0,2nd stage)
= + E0ods(POIo,t) *{ 1 + 3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 + 21/8*β^4*sin(Aθpc(POIo(t),t=0))^4 }
- E0ods(POIo,t)*λ(Vons(PART)) *{ 1 + β^2*sin(Aθpc(POIo(t),t=0))^2 }
/* 16Oct2019 As 4-35 failed, there is no sense going any further, as I've done similar derivations many times befoire, and this isn't working [functionally, coefficients]!!!
# enddoc
/*_endCmd
**********************************************************
/*------> (4-37) Er and the binomial series, leading to the relativistic correction factor
27Sep2019 - did not finish - revamp in next attempt with updated previous equations 4-[32 to 36]
27Sep2019 using updated previous equations 4-[32 to 36]
02Oct2019 add [K_1st, K_2nd] terms
after fixing ∂[∂(t): Rpcs(POIo(t),t)^(-b)*sin(Aθpc(POIo(t),t))^a]
fix ∂[∂(t): E0ods(POIo,t) *sin(Aθpc(POIo(t),t))^a]
11Oct2019 use corrected form of EIods(POIo,t,2nd stage) with "+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}"
Approach for 22Aug2019, 03Sep2019 after correcting recent errors in 04_35 :
1. Don't track Lucas too closely - he has wrong coefficients
2. Iterations pass through same operations as (4-32) through (4-36)
3. Howells "dp[dt : E0ods(POIo,t)] = dp[dt : E0pds(POIo,t)] - see proper E0odv(POIo,t) vector approach" from "Howell - Background math for Lucas Universal Force, Chapter 4.odt"
I was very lazy with symbols -0 many improper uses!!! Must clean up thre t=0 notational mess for [integrals, derivatives]
/*/*$ cat >>"$p_augmented" "$d_augment""cos - 1 $cos_inclusion, iterative, non-feedback/04_37 work.txt"
/*-------------------------------------------
/* Targeted results /%
(mathL)/* theoretical target - binomial series /%
EIods(POIo,t,4th stage)
= + E0pds(POIp) *{ 3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 + 3/8*β^4*sin(Aθpc(POIo(t),t=0))^4 + -1/16*β^6*sin(Aθpc(POIo(t),t=0))^6 + 3/128*β^8*sin(Aθpc(POIo(t),t=0))^8}
- E0pds(POIp)*λ(Vons(PART)) *{1 + 3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 + 3/8*β^4*sin(Aθpc(POIo(t),t=0))^4 + -1/16*β^6*sin(Aθpc(POIo(t),t=0))^6}
(endMath)
(mathL)
binomial series = 1 3/2 3/8 -1/16 3/128 -3/256 7/1024
Lucas = 1 3/2 15/8 35/16
12Oct2019
ETpds(POIp) = 1 3/2 21/8 35/8 455/64
ETpds(POIp)*λ(Vons(PART)) = 1 1 5/4 5/3
(endMath)
/* 16Oct2019 As 4-35 failed, there is no sense going any further, as I've done similar derivations many times befoire, and this isn't working [functionally, coefficients]!!!
/*_endCmd
enddoc