/* 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*(vr´)/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