/* 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)
/* 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 :
/%
2) ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*∂[∂(t): EIods(POIo,t)]*sin(Aθpc(POIo(t),t)) ]
/* But all that I really need to do is to take the derivative of (1) and directly put that into the integral in (1)
Taking the partial derivative of EIods(POIo,t) :
/%
3) ∂[∂(t): EIods(POIo,t,1st stage)]
= + ∂[∂(t): 3/2*β^2 *Q(PART)*Rocs(POIo)^3 *Rpcs(POIo(t),t=0)^(-5) * sin(Aθpc(POIo(t),t=0))^2 ]
+ ∂[∂(t): 3 *β^2 *Q(PART)*Rocs(POIo)^2 *Rpcs(POIo(t),t=0)^(-5)*Vons(PART)*t*(cos(Aθpc(POIo(t),t=0)) - 1) ]
- ∂[∂(t): λ(Vons(PART)) *Q(PART) *Rpcs(POIo(t),t=0)^(-2) ]
+ ∂[∂(t): f_sphereCapSurf(EIods(POIo,t,0th stage)) ]
/* Putting (3) into (1) Yields : /%
4) EIods(POIo,t=0)
= + K0 + K1 + K2
+ β^1*Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
*{
+ ∂[∂(t): 3/2*β^2 *Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t=0)^(-5) * sin(Aθpc(POIo(t),t=0))^2 ]
+ ∂[∂(t): 3 *β^2 *Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(-5)*Vons(PART)*t*(cos(Aθpc(POIo(t),t=0)) - 1) ]
- ∂[∂(t): λ(Vons(PART)) *Q(PART) *Rpcs(POIo(t),t=0)^(-2) ]
+ ∂[∂(t): f_sphereCapSurf(EIods(POIo,t,0th stage)) ]
}
]
/* Lucas instructs iterative substitutions for Ei(ro - vo*t,t) at t=0 implicitly, and dropping v*t*(cosO - 1) terms as we go.
But why is it still in Equation (4-32)??
/* using /%
t=0
∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] = 0
2590:(mathH) ∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] = 0
2304:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-5)] = 7*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6)
/*_file_insert_path "$d_Lucas""relativistic factor, intermediate symbols.txt"
/* Also, as the f_sphereCapSurf(EIods(POIo,t)) term is dropped after integration, it is convenient to show it separately
This also makes the meaning of the
/%
5) EIods(POIo,t=0,2nd stage)
= + K_1st
+ β^1*Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
*{
+ ∂[∂(t): 3/2*β^2 *Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t=0)^(-5) *sin(Aθpc(POIo(t),t=0))^2 ]
- ∂[∂(t): λ(Vons(PART)) *Q(PART) *Rpcs(POIo(t),t=0)^(-2) ]
}
]
+ f_sphereCapSurf{∂[∂(t): f_sphereCapSurf(EIods(POIo,t,0th stage))]}
/*_file_insert_path "$d_Lucas""relativistic factor, intermediate symbols.txt"
/* |--???--> (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)
/$L Eis(r - v*t,t) APPLY |t=0 TO EACH TERM
= K0 + K2
+ β*rs^2*∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t): K0 + K2])
+ β*rs^2*∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t): K3])
/* 03Sep2019 This is old!
/$H Eis(r - v*t,t) APPLY |t=0 TO EACH TERM
= K0 + K2
+ β*rs^2*∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t): K0 + K2])
+ β*rs^2*∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t): K3])
/* OK - works great by using a blend of Lucas & Howell expressions for (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 /%
EIods(POIo,t=0,2nd stage)
= + K_1st
+ β^1*Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
*{
+ ∂[∂(t): 3/2*β^2 *Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t=0)^(-5) *sin(Aθpc(POIo(t),t=0))^2 ]
- ∂[∂(t): λ(Vons(PART)) *Q(PART) *Rpcs(POIo(t),t=0)^(-2) ]
}
]
+ f_sphereCapSurf{∂[∂(t): f_sphereCapSurf(EIods(POIo,t,0th stage))]}
/*_file_insert_path "$d_Lucas""relativistic factor, intermediate symbols.txt"