Following 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. Presumably, at t=0 cosθ = 1, so (cosO - 1)|t=0 = 0 Notice that the expression variables are ALL scalar 04_30rev4 Lorentz force - Faradays_law_integrated /* Howells Flat-Liner Notation (HFLN) From (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)) /*_file_insert_path "$d_Lucas""relativistic factor, intermediate symbols.txt" /**************************************************************************** >>>>>>>>> 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) /*_file_insert_path "$d_Lucas""relativistic factor, intermediate symbols.txt" /* 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" !?!? # enddoc