/*--------------------------------------------------- /* Stage 3 iteration /* Restating iterative solution : (mathL)/* generative form /% EIods(POIo,t,3rd stage) = K_1st + f_sphereCapSurf(EIods(POIo,t,2nd stage)) (endMath) /* using /% 3023:(mathL)/* differentiable form /% EIods(POIo,t,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) - 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))]} (endMath) 1) EIods(POIo,t,3rd stage) = + K_1st + f_sphereCapSurf ( + 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) - 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))]} ) = + K_1st + Q(PART)*f_sphereCapSurf ( 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) - 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{f_sphereCapSurf(EIods(POIo,t,0th stage))}} /* using /% 1042:(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]] 3) EIods(POIo,t,3rd stage) = + K_1st + Q(PART)*β*Rocs(POIo)^2*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t)) 3a) *∂[∂(t): ( 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) - 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{f_sphereCapSurf(EIods(POIo,t,0th stage))}} /* +-----+ Looking at (3a) "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" [c, β, λ(Vons(PART)), Q(PART), Rocs(POIo), Vons(PART)] - are ALWAYS constants in Chapter 4 [Rpcs(POIo(t),t), E0ods(POIo,t)] - are NOT constants wrt ∂[∂(t):, ARE constants wrt ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): /% 3b) *( + 3/2 *β^2*Rocs(POIo)^3 *∂[∂(t): sin(Aθpc(POIo(t),t=0))^2 *Rpcs(POIo(t),t=0)^(-5)] + 21/8 *β^4*Rocs(POIo)^4 *∂[∂(t): sin(Aθpc(POIo(t),t=0))^4 *Rpcs(POIo(t),t=0)^(-6)] - λ(Vons(PART)) *1 *∂[∂(t): *Rpcs(POIo(t),t=0)^(-2)] - λ(Vons(PART)) *1 *β^2*Rocs(POIo) *∂[∂(t): sin(Aθpc(POIo(t),t=0))^2 *Rpcs(POIo(t),t=0)^(-3)] ) /* using /% 2286:(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) 2300:(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) 2263:(mathH) ∂[∂(t): Rpcs(POIo(t),t)^(-2)] = 2*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) 2282:(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) 3c) *( + 3/2 *β^2*Rocs(POIo)^3 * 7*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(-6) + 21/8 *β^4*Rocs(POIo)^4 *10*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(-7) - λ(Vons(PART)) * 2*Vons(PART) *cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(-3) - λ(Vons(PART)) *β^2*Rocs(POIo) * 5*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(-4) ) /* multiply numbers /% 3d) *( + 21/2 *β^2*Rocs(POIo)^3 *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(-6) + 105/4 *β^4*Rocs(POIo)^4 *Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(-7) - λ(Vons(PART)) *2 *Vons(PART) *cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(-3) - λ(Vons(PART)) *5 *β^2*Rocs(POIo) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(-4) ) /* substitute (3d) for (3a) /% 4) EIods(POIo,t,3rd stage) = + K_1st 4a) + Q(PART)*β*Rocs(POIo)^2*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t)) *( + 21/2 *β^2*Rocs(POIo)^3 *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(-6) + 105/4 *β^4*Rocs(POIo)^4 *Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(-7) - λ(Vons(PART)) *2 *Vons(PART) *cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(-3) - λ(Vons(PART)) *5 *β^2*Rocs(POIo) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(-4) ) + f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}} /* +-----+ Looking at (4a) "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" [c, β, λ(Vons(PART)), Q(PART), Rocs(POIo), Vons(PART)] - are ALWAYS constants in Chapter 4 [Rpcs(POIo(t),t), E0ods(POIo,t)] - are NOT constants wrt ∂[∂(t):, ARE constants wrt ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): /% 4b) + Q(PART)*β*Rocs(POIo)^2*1/c/Rocs(POIo) *( + 21/2 *β^2*Rocs(POIo)^3 *Vons(PART)*Rpcs(POIo(t),t)^(-6) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t)) + 105/4 *β^4*Rocs(POIo)^4 *Vons(PART)*Rpcs(POIo(t),t)^(-7) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t)) - λ(Vons(PART)) *2 *Vons(PART)*Rpcs(POIo(t),t)^(-3) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t)) *cos(Aθpc(POIo(t),t)) - λ(Vons(PART)) *5 *β^2*Rocs(POIo) *Vons(PART)*Rpcs(POIo(t),t)^(-4) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t)) ) /* collect powers of [sin(Aθpc(POIo(t),t)), Rocs(POIo)], move Rocs(POIo)into each line, extract Vons(PART) from each line /% 4c) + Q(PART)*β*Vons(PART)/c *( + 21/2 *β^2*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(-6) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^3 *cos(Aθpc(POIo(t),t)) + 105/4 *β^4*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(-7) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^5 *cos(Aθpc(POIo(t),t)) - λ(Vons(PART)) *2 *Rocs(POIo)^1 *Rpcs(POIo(t),t)^(-3) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^1 *cos(Aθpc(POIo(t),t)) - λ(Vons(PART)) *5 *β^2*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-4) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^3 *cos(Aθpc(POIo(t),t)) ) /* using /% 362:(mathL) β = Vons(PART)/c 4d) + Q(PART)*β^2 *( + 21/2 *β^2*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(-6) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^3 *cos(Aθpc(POIo(t),t)) + 105/4 *β^4*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(-7) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^5 *cos(Aθpc(POIo(t),t)) - λ(Vons(PART)) *2 *Rocs(POIo)^1 *Rpcs(POIo(t),t)^(-3) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^1 *cos(Aθpc(POIo(t),t)) - λ(Vons(PART)) *5 *β^2*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-4) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^3 *cos(Aθpc(POIo(t),t)) ) /* collect powers of β /% 4e) + Q(PART) *( + 21/2 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(-6) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^3 *cos(Aθpc(POIo(t),t)) + 105/4 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(-7) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^5 *cos(Aθpc(POIo(t),t)) - λ(Vons(PART)) *2 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(-3) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^1 *cos(Aθpc(POIo(t),t)) - λ(Vons(PART)) *5 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-4) *∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^3 *cos(Aθpc(POIo(t),t)) ) /* using /% 3099:(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 3103:(mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^5] = sin(Aθpc(POIo(t),t=0))^6/6 3095:(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 3099:(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 4f) + Q(PART) *( + 21/2 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(-6) *sin(Aθpc(POIo(t),t=0))^4/4 + 105/4 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(-7) *sin(Aθpc(POIo(t),t=0))^6/6 - λ(Vons(PART)) *2 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(-3) *sin(Aθpc(POIo(t),t=0))^2/2 - λ(Vons(PART)) *5 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-4) *sin(Aθpc(POIo(t),t=0))^4/4 /* collect numbers /% 4g) + Q(PART) *( + 21/8 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(-6) *sin(Aθpc(POIo(t),t=0))^4 + 35/8 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(-7) *sin(Aθpc(POIo(t),t=0))^6 - λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(-3) *sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-4) *sin(Aθpc(POIo(t),t=0))^4 ) /* insert (4g) in place of (4a), this can be used for t=0 form /% 5) EIods(POIo,t,3rd stage) = + K_1st + Q(PART) *( + 21/8 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(-6) *sin(Aθpc(POIo(t),t=0))^4 + 35/8 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(-7) *sin(Aθpc(POIo(t),t=0))^6 - λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(-3) *sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-4) *sin(Aθpc(POIo(t),t=0))^4 ) + f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}} /* drop "=0" qualifier to get differentiable form /% /* using in (5) /% 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,3rd stage) = + 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) ) + Q(PART) *( + 21/8 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^4 + 35/8 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(-7)*sin(Aθpc(POIo(t),t))^6 - λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t))^2 - λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^4 ) + f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}} 7) EIods(POIo,t,3rd stage) = + Q(PART) *( + 3/2 *β^2*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2 + 21/8 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^4 + 35/8 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(-7)*sin(Aθpc(POIo(t),t))^6 - λ(Vons(PART)) *1 *Rpcs(POIo(t),t)^(-2) - λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t))^2 - λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^4 ) + f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}} /* remove "=0" from t to get : (mathL)/* differentiable form /% EIods(POIo,t,3rd stage) = + Q(PART) *( + 3/2 *β^2*Rocs(POIo)^3 *sin(Aθpc(POIo(t),t))^2 *Rpcs(POIo(t),t)^(-5) + 21/8 *β^4*Rocs(POIo)^4 *sin(Aθpc(POIo(t),t))^4 *Rpcs(POIo(t),t)^(-6) + 35/8 *β^6*Rocs(POIo)^5 *sin(Aθpc(POIo(t),t))^6 *Rpcs(POIo(t),t)^(-7) - λ(Vons(PART)) *1 *Rpcs(POIo(t),t)^(-2) - λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *sin(Aθpc(POIo(t),t))^2 *Rpcs(POIo(t),t)^(-3) - λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *sin(Aθpc(POIo(t),t))^4 *Rpcs(POIo(t),t)^(-4) ) + f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}} (endMath) /* post-[differentiation, integration] form /% /* 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" /% 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) /* use this in (7), careful to put t=0 /% 7) EIods(POIo,t,3rd stage) = + Q(PART) *( + 3/2 *β^2 *Rpcs(POIo(t),t=0)^(-2)*sin(Aθpc(POIo(t),t=0))^2 + 21/8 *β^4 *Rpcs(POIo(t),t=0)^(-2)*sin(Aθpc(POIo(t),t=0))^4 + 35/8 *β^6 *Rpcs(POIo(t),t=0)^(-2)*sin(Aθpc(POIo(t),t=0))^6 - λ(Vons(PART)) *1 *Rpcs(POIo(t),t=0)^(-2) - λ(Vons(PART)) *1 *β^2 *Rpcs(POIo(t),t=0)^(-2)*sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART)) *5/4 *β^4 *Rpcs(POIo(t),t=0)^(-2)*sin(Aθpc(POIo(t),t=0))^4 ) + f_sphereCapSurf{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 + 35/8 *β^6 *sin(Aθpc(POIo(t),t=0))^6 - λ(Vons(PART)) *1 - λ(Vons(PART)) *1 *β^2 *sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART)) *5/4 *β^4 *sin(Aθpc(POIo(t),t=0))^4 ) + f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}} /* using /% 1174:(mathH) E0pds(POIp) = Q(PART)/Rpcs(POIp)^2 8) EIods(POIo,t=0,3rd stage) = + E0pds(POIp) *3/2 *β^2*sin(Aθpc(POIo(t),t=0))^2 + E0pds(POIp) *21/8 *β^4*sin(Aθpc(POIo(t),t=0))^4 + E0pds(POIp) *35/8 *β^6*sin(Aθpc(POIo(t),t=0))^6 - E0pds(POIp)*λ(Vons(PART)) *1 - E0pds(POIp)*λ(Vons(PART)) *1 *β^2*sin(Aθpc(POIo(t),t=0))^2 - E0pds(POIp)*λ(Vons(PART)) *5/4 *β^4*sin(Aθpc(POIo(t),t=0))^4 + f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}} (mathL)/* HIGHLY restrictive conditions : t=0, RFo(t=0) = Rfp(t=0), Rocs(POIo) = Rpcs(POIo(t),t=0) /% EIods(POIo,t=0,3rd stage) = + E0pds(POIp) *{ 3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 + 21/8*β^4*sin(Aθpc(POIo(t),t=0))^4 + 35/8*β^6*sin(Aθpc(POIo(t),t=0))^6} - E0pds(POIp)*λ(Vons(PART)) *{1 + β^2*sin(Aθpc(POIo(t),t=0))^2 + 5/4 *β^4*sin(Aθpc(POIo(t),t=0))^4} (endMath) /*_file_insert_path "$d_Lucas""relativistic factor, intermediate symbols.txt" /*--------------------------------------------------- /* Stage 4 iteration /* Start with iteration 3 of 11Oct2019 /% 3284:(mathL)/* differentiable form /% EIods(POIo,t,3rd stage) = + Q(PART) *( + 3/2 *β^2*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2 + 21/8 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^4 + 35/8 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(-7)*sin(Aθpc(POIo(t),t))^6 - λ(Vons(PART)) *1 *Rpcs(POIo(t),t)^(-2) - λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t))^2 - λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^4 ) + f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}} /* see "textIn - relativistic factor, intermediate symbols, restrictive conditions.txt" Restating iterative solution : /% (mathL)/* generative form /% EIods(POIo,t,4th stage) = K_1st + f_sphereCapSurf(EIods(POIo,t,3rd stage)) (endMath) 1) EIods(POIo,t,4th stage) = + K_1st + f_sphereCapSurf ( + Q(PART) *( + 3/2 *β^2*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2 + 21/8 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^4 + 35/8 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(-7)*sin(Aθpc(POIo(t),t))^6 - λ(Vons(PART)) *1 *Rpcs(POIo(t),t)^(-2) - λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t))^2 - λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^4 ) + f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}} ) = + K_1st + Q(PART)*f_sphereCapSurf ( + 3/2 *β^2*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2 + 21/8 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^4 + 35/8 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(-7)*sin(Aθpc(POIo(t),t))^6 - λ(Vons(PART)) *1 *Rpcs(POIo(t),t)^(-2) - λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t))^2 - λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^4 ) + f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}}} /* using /% 1042:(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]] 2) EIods(POIo,t,4th stage) = + K_1st + Q(PART)*β*Rocs(POIo)^2*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t)) 2a) *∂[∂(t): ( + 3/2 *β^2*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2 + 21/8 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^4 + 35/8 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(-7)*sin(Aθpc(POIo(t),t))^6 - λ(Vons(PART)) *1 *Rpcs(POIo(t),t)^(-2) - λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t))^2 - λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^4 ) ] + f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}}} /* +-----+ Looking at (3a) "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" [c, β, λ(Vons(PART)), Q(PART), Rocs(POIo), Vons(PART)] - are ALWAYS constants in Chapter 4 [Rpcs(POIo(t),t), E0ods(POIo,t)] - are NOT constants wrt ∂[∂(t):, ARE constants wrt ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): /% 2b) ( + 3/2 *β^2*Rocs(POIo)^3 *∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-5)] + 21/8 *β^4*Rocs(POIo)^4 *∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-6)] + 35/8 *β^6*Rocs(POIo)^5 *∂[∂(t): sin(Aθpc(POIo(t),t))^6*Rpcs(POIo(t),t)^(-7)] - λ(Vons(PART)) *1 *∂[∂(t): *Rpcs(POIo(t),t)^(-2)] - λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-3)] - λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-4)] ) /* using /% 2286:(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) 2300:(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) 2336:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^6*Rpcs(POIo(t),t)^(-7)] = 13*Vons(PART)*sin(Aθpc(POIo(t),t))^6*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-8) 2263:(mathH) ∂[∂(t): Rpcs(POIo(t),t)^(-2)] = 2*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) 2282:(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) 2314:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-4)] = 8*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5) 2c) ( + 3/2 *β^2*Rocs(POIo)^3 * 7*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) + 21/8 *β^4*Rocs(POIo)^4 *10*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) + 35/8 *β^6*Rocs(POIo)^5 *13*Vons(PART)*sin(Aθpc(POIo(t),t))^6*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-8) - λ(Vons(PART)) *1 * 2*Vons(PART) *cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) - λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 * 5*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) - λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 * 8*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5) ) /* multiply numbers /% 2d) ( + 21/2 *β^2*Rocs(POIo)^3 *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) + 105/4 *β^4*Rocs(POIo)^4 *Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) + 455/8 *β^6*Rocs(POIo)^5 *Vons(PART)*sin(Aθpc(POIo(t),t))^6*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-8) - λ(Vons(PART)) *2 *Vons(PART) *cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) - λ(Vons(PART)) *5 *β^2*Rocs(POIo)^1 *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) - λ(Vons(PART)) *10 *β^4*Rocs(POIo)^2 *Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5) ) /* substitute (2d) for (2a) /% 3) EIods(POIo,t,4th stage) = + K_1st 3a) + Q(PART)*β*Rocs(POIo)^2*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t)) *( + 21/2 *β^2*Rocs(POIo)^3 *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) + 105/4 *β^4*Rocs(POIo)^4 *Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) + 455/8 *β^6*Rocs(POIo)^5 *Vons(PART)*sin(Aθpc(POIo(t),t))^6*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-8) - λ(Vons(PART)) *2 *Vons(PART) *cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) - λ(Vons(PART)) *5 *β^2*Rocs(POIo)^1 *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) - λ(Vons(PART)) *10 *β^4*Rocs(POIo)^2 *Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5) ) + f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}}} /* +-----+ Looking at (3a) "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" [c, β, λ(Vons(PART)), Q(PART), Rocs(POIo), Vons(PART)] - are ALWAYS constants in Chapter 4 [Rpcs(POIo(t),t), E0ods(POIo,t)] - are NOT constants wrt ∂[∂(t):, ARE constants wrt ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): /% 3b) + Q(PART)*β*Rocs(POIo)^2 *( + 21/2 *β^2*Rocs(POIo)^3 *Vons(PART)*Rpcs(POIo(t),t)^(-6)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))] + 105/4 *β^4*Rocs(POIo)^4 *Vons(PART)*Rpcs(POIo(t),t)^(-7)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))] + 455/8 *β^6*Rocs(POIo)^5 *Vons(PART)*Rpcs(POIo(t),t)^(-8)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^6*cos(Aθpc(POIo(t),t))] - λ(Vons(PART)) *2 *Vons(PART)*Rpcs(POIo(t),t)^(-3)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t)) *cos(Aθpc(POIo(t),t))] - λ(Vons(PART)) *5 *β^2*Rocs(POIo)^1 *Vons(PART)*Rpcs(POIo(t),t)^(-4)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))] - λ(Vons(PART)) *10 *β^4*Rocs(POIo)^2 *Vons(PART)*Rpcs(POIo(t),t)^(-5)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))] ) /* collect powers of [sin(Aθpc(POIo(t),t)), Rocs(POIo)], move Rocs(POIo)into each line, extract Vons(PART) from each line /% 3c) + Q(PART)*β*Vons(PART)/c *( + 21/2 *β^2*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(-6)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))] + 105/4 *β^4*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(-7)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^5*cos(Aθpc(POIo(t),t))] + 455/8 *β^6*Rocs(POIo)^6 *Rpcs(POIo(t),t)^(-8)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^7*cos(Aθpc(POIo(t),t))] - λ(Vons(PART)) *2 *Rocs(POIo)^1 *Rpcs(POIo(t),t)^(-3)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^1*cos(Aθpc(POIo(t),t))] - λ(Vons(PART)) *5 *β^2*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-4)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))] - λ(Vons(PART)) *10 *β^4*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-5)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^5*cos(Aθpc(POIo(t),t))] ) /* using /% 362:(mathL) β = Vons(PART)/c 3d) + Q(PART)*β^2 *( + 21/2 *β^2*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(-6)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))] + 105/4 *β^4*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(-7)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^5*cos(Aθpc(POIo(t),t))] + 455/8 *β^6*Rocs(POIo)^6 *Rpcs(POIo(t),t)^(-8)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^7*cos(Aθpc(POIo(t),t))] - λ(Vons(PART)) *2 *Rocs(POIo)^1 *Rpcs(POIo(t),t)^(-3)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^1*cos(Aθpc(POIo(t),t))] - λ(Vons(PART)) *5 *β^2*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-4)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))] - λ(Vons(PART)) *10 *β^4*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-5)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^5*cos(Aθpc(POIo(t),t))] ) /* collect powers of β /% 3e) + Q(PART) *( + 21/2 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(-6)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))] + 105/4 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(-7)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^5*cos(Aθpc(POIo(t),t))] + 455/8 *β^8*Rocs(POIo)^6 *Rpcs(POIo(t),t)^(-8)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^7*cos(Aθpc(POIo(t),t))] - λ(Vons(PART)) *2 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(-3)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^1*cos(Aθpc(POIo(t),t))] - λ(Vons(PART)) *5 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-4)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))] - λ(Vons(PART)) *10 *β^6*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-5)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))^5*cos(Aθpc(POIo(t),t))] ) /* using /% 3099:(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 3103:(mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^5] = sin(Aθpc(POIo(t),t=0))^6/6 3187:(mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^7] = sin(Aθpc(POIo(t),t=0))^8/8 3095:(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 3099:(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 3183:(mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^5] = sin(Aθpc(POIo(t),t=0))^6/6 3f) + Q(PART) *( + 21/2 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4/4 + 105/4 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(-7)*sin(Aθpc(POIo(t),t=0))^6/6 + 455/8 *β^8*Rocs(POIo)^6 *Rpcs(POIo(t),t)^(-8)*sin(Aθpc(POIo(t),t=0))^8/8 - λ(Vons(PART)) *2 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t=0))^2/2 - λ(Vons(PART)) *5 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t=0))^4/4 - λ(Vons(PART)) *10 *β^6*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t=0))^6/6 ) /* collect numbers /% 3g) + Q(PART) *( + 21/8 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4 + 35/8 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(-7)*sin(Aθpc(POIo(t),t=0))^6 + 455/64*β^8*Rocs(POIo)^6 *Rpcs(POIo(t),t)^(-8)*sin(Aθpc(POIo(t),t=0))^8 - λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t=0))^4 - λ(Vons(PART)) *5/3 *β^6*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t=0))^6 ) /* insert (3g) in place of (3a), this can be used for t=0 form /% 4) EIods(POIo,t,4th stage) = + K_1st + Q(PART) *( + 21/8 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4 + 35/8 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(-7)*sin(Aθpc(POIo(t),t=0))^6 + 455/64*β^8*Rocs(POIo)^6 *Rpcs(POIo(t),t)^(-8)*sin(Aθpc(POIo(t),t=0))^8 - λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t=0))^4 - λ(Vons(PART)) *5/3 *β^6*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t=0))^6 ) + f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}}} /* drop "=0" qualifier to get differentiable form /% /* 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) ) 5) EIods(POIo,t,4th stage) = + 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) ) + Q(PART) *( + 21/8 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4 + 35/8 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(-7)*sin(Aθpc(POIo(t),t=0))^6 + 455/64*β^8*Rocs(POIo)^6 *Rpcs(POIo(t),t)^(-8)*sin(Aθpc(POIo(t),t=0))^8 - λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t=0))^4 - λ(Vons(PART)) *5/3 *β^6*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t=0))^6 ) + f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}}} 6) EIods(POIo,t,4th stage) = + Q(PART) *( + 3/2 *β^2*Rocs(POIo)^3 *Rpcs(POIo(t),t=0)^(-5)*sin(Aθpc(POIo(t),t))^2 + 21/8 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t=0)^(-6)*sin(Aθpc(POIo(t),t=0))^4 + 35/8 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t=0)^(-7)*sin(Aθpc(POIo(t),t=0))^6 + 455/64*β^8*Rocs(POIo)^6 *Rpcs(POIo(t),t=0)^(-8)*sin(Aθpc(POIo(t),t=0))^8 - λ(Vons(PART)) *1 *Rpcs(POIo(t),t=0)^(-2) - λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t=0)^(-3)*sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t=0)^(-4)*sin(Aθpc(POIo(t),t=0))^4 - λ(Vons(PART)) *5/3 *β^6*Rocs(POIo)^3 *Rpcs(POIo(t),t=0)^(-5)*sin(Aθpc(POIo(t),t=0))^6 ) + f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}}} /* remove "=0" from t to get : (mathL)/* differentiable form /% EIods(POIo,t,4th stage) = + Q(PART) *( + 3/2 *β^2*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2 + 21/8 *β^4*Rocs(POIo)^4 *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^4 + 35/8 *β^6*Rocs(POIo)^5 *Rpcs(POIo(t),t)^(-7)*sin(Aθpc(POIo(t),t))^6 + 455/64*β^8*Rocs(POIo)^6 *Rpcs(POIo(t),t)^(-8)*sin(Aθpc(POIo(t),t))^8 - λ(Vons(PART)) *1 *Rpcs(POIo(t),t)^(-2) - λ(Vons(PART)) *1 *β^2*Rocs(POIo)^1 *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t))^2 - λ(Vons(PART)) *5/4 *β^4*Rocs(POIo)^2 *Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^4 - λ(Vons(PART)) *5/3 *β^6*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^6 ) + f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}}}} (endMath) /* post-[differentiation, integration] form /% /* HIGHLY restrictive conditions : t=0, RFo(t=0) = Rfp(t=0), Rocs(POIo) = Rpcs(POIo(t),t=0) /% 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) /* use this in (6), careful to put t=0 /% 7) EIods(POIo,t,4th stage) = + Q(PART) *( + 3/2 *β^2 *Rpcs(POIo(t),t=0)^(-2)*sin(Aθpc(POIo(t),t=0))^2 + 21/8 *β^4 *Rpcs(POIo(t),t=0)^(-2)*sin(Aθpc(POIo(t),t=0))^4 + 35/8 *β^6 *Rpcs(POIo(t),t=0)^(-2)*sin(Aθpc(POIo(t),t=0))^6 + 455/64*β^8 *Rpcs(POIo(t),t=0)^(-2)*sin(Aθpc(POIo(t),t=0))^8 - λ(Vons(PART)) *1 *Rpcs(POIo(t),t=0)^(-2) - λ(Vons(PART)) *1 *β^2 *Rpcs(POIo(t),t=0)^(-2)*sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART)) *5/4 *β^4 *Rpcs(POIo(t),t=0)^(-2)*sin(Aθpc(POIo(t),t=0))^4 - λ(Vons(PART)) *5/3 *β^6 *Rpcs(POIo(t),t=0)^(-2)*sin(Aθpc(POIo(t),t=0))^6 ) = + 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 + 35/8 *β^6 *sin(Aθpc(POIo(t),t=0))^6 + 455/64*β^8 *sin(Aθpc(POIo(t),t=0))^8 - λ(Vons(PART)) *1 - λ(Vons(PART)) *1 *β^2 *sin(Aθpc(POIo(t),t=0))^2 - λ(Vons(PART)) *5/4 *β^4 *sin(Aθpc(POIo(t),t=0))^4 - λ(Vons(PART)) *5/3 *β^6 *sin(Aθpc(POIo(t),t=0))^6 ) /* using /% 1174:(mathH) E0pds(POIp) = Q(PART)/Rpcs(POIp)^2 8) EIods(POIo,t,4th stage) = + E0pds(POIp) *3/2 *β^2 *sin(Aθpc(POIo(t),t=0))^2 + E0pds(POIp) *21/8 *β^4 *sin(Aθpc(POIo(t),t=0))^4 + E0pds(POIp) *35/8 *β^6 *sin(Aθpc(POIo(t),t=0))^6 + E0pds(POIp) *455/64 *β^8 *sin(Aθpc(POIo(t),t=0))^8 - E0pds(POIp)*λ(Vons(PART)) *1 - E0pds(POIp)*λ(Vons(PART)) *1 *β^2 *sin(Aθpc(POIo(t),t=0))^2 - E0pds(POIp)*λ(Vons(PART)) *5/4 *β^4 *sin(Aθpc(POIo(t),t=0))^4 - E0pds(POIp)*λ(Vons(PART)) *5/3 *β^6 *sin(Aθpc(POIo(t),t=0))^6 (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,4th stage) = + E0pds(POIp) *{ 3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 + 21/8*β^4*sin(Aθpc(POIo(t),t=0))^4 + 35/8*β^6*sin(Aθpc(POIo(t),t=0))^6 + 455/64*β^8*sin(Aθpc(POIo(t),t=0))^8} - E0pds(POIp)*λ(Vons(PART)) *{1 + β^2*sin(Aθpc(POIo(t),t=0))^2 + 5/4 *β^4*sin(Aθpc(POIo(t),t=0))^4 + 5/3 *β^6*sin(Aθpc(POIo(t),t=0))^6} (endMath) 12Oct2019 This isn't working - the numbers are blowing up!! Seems to be missing negative terms... However, at least the powers of the terms are OK now, which is a HUGE improvement! /*_file_insert_path "$d_Lucas""relativistic factor, restrictive conditions.txt" # enddoc