/* 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