FORM 4.0 (Jun 30 2012) 64-bits Run: Sat Aug 3 17:46:03 2013 * Title: SMALL CHARGE: * proper elastic potential energy density * File: elasemPE.frm * Author: David N. Williams * License: Creative Commons Attribution-Share Alike * Started: July 26, 2012 * Revised: July 26-30, 2012 * August 1-3, 2012 * October 17, 2012 improved comments * February 4, 2013 fixed sign of dR.dF/dzet * March 25, 2013 renamed F as U * August 1, 2013 felas output * August 2, 2013 renamed kappa and zeta as kap and zet * August 3, 2013 renamed [dL.dU/dzet] and [dR.dU/dzet] as * [d2U/dalp.dzet] and [d2U/dzet.dalp] * * Copyright (C) 2012, 2013 David N. Williams * * This work is licensed under the Creative Commons Attribution- * Share Alike 2.5 License. To view a copy of this license, visit * http://creativecommons.org/licenses/by-sa/2.5/ or send a letter * to Creative Commons, 543 Howard Street, 5th Floor, San * Francisco, California, 94105, USA. * * Purely cosmetic changes may not be reflected in the revision * dates above. * * Input: elasforce.sav, lfsing.sav * Output: elasemPE.log * felas in elasenPE.sav #if 0 This program identifies a proper elastic potential energy density for which the negative divergence of the elastic energy-momentum tensor cancels the ℓ⁻¹ term in the retarded/advanced Lorentz force density for a spherically symmetric charge with small radius ℓ, leaving part of the ℓ⁻² term, and adding an extra ℓ⁻¹ term proportion to ∂ρ₀/∂αⁱ, where ρ₀ = ρ₀(α) is the proper charge density. In the following, the Lorentz force density lfsing, loaded from lfsing.sav, is missing an overall factor ρ₀/Jγ, where Jγ is an invariant, four-dimensional Jacobian. The model for the proper potential energy function U is missing the same factor, which has the effect that the model elastic force density felas lacks the extra term with ∂ρ₀/∂αⁱ mentioned above. These omissions are easily corrected, and that has been done in subsequent calculations where it matters. The U term in felas is felas(mu) = - a(mu)*c^-2*U + ... The corresponding a(mu) term in lfsing, without the overall ρ₀/Jγ, is lfsing(mu) = + eps^-1*snorm^-1 * ( - 1/2*a(mu)*c^-2 + ... Hence the model for U without ρ₀/Jγ is U = -1/2*eps^-1*snorm^-1 Note that felas = -divE. ζᵢⱼ = -ηᵢ⋅ηⱼ = -∂ᵢy⋅∂ⱼy + u⋅∂ᵢy u⋅∂ⱼy/c² Substituting for U and its derivatives generates dot products ∂ᵢy⋅∂ⱼy and ηᵢ⋅ηⱼ, which can be simplified with: ηᵢ⋅ηⱼ = ∂ᵢy⋅∂ⱼy - u⋅∂ᵢy u⋅∂ⱼy/c² = -ζᵢⱼ ⇒ ∂ᵢy⋅∂ⱼy = u⋅∂ᵢy u⋅∂ⱼy/c² - ζᵢⱼ #endif *** DECLARATIONS OFF statistics; S c,U,p,L,T; I mu,nu,la,i,i1,i2,i3,i4; V y,u,a,n; T d1y,d2y,d3y,d1u,d2u,d1a; T eta; CF [DL.dU/dzet],[DR.dU/dzet],[dL.dU/dzet],[dR.dU/dzet]; CF [DL.dU/dzet],[DR.dU/dzet],[d2U/dalp.dzet],[d2U/dzet.dalp]; T [dU/dzet],zetdot(symmetric),Deta,deta; T dkap,dzet; CF kap,kapdot,[d2U/dzet2]; * extra declarations from lfsing.frm S eps,snorm; Load elasforce.sav; divE loaded Load lfsing.sav; lfsing loaded *** DEBUGGING #procedure try .sort B eps,snorm; print +s; .end #endprocedure *** MODULES G felas(mu) = -divE(mu); Id U = -1/2*eps^-1*snorm^-1; Id [dU/dzet](i1?,i2?) = 1/4*eps^-1*snorm^-3*n(i1)*n(i2); Id [d2U/dzet2](i1?,i2?,i3?,i4?) = -3/8*eps^-1*snorm^-5 *n(i1)*n(i2)*n(i3)*n(i4); Id [d2U/dalp.dzet](i1?) = [d2U/dzet.dalp](i1); Id [d2U/dzet.dalp](i1?) = -1/4*eps^-2*snorm^-3*n(i1); * ∂ᵢy⋅∂ⱼy = u⋅∂ᵢy u⋅∂ⱼy/c² - ζᵢⱼ * Id d1y(mu?,n)*d1y(mu?,n) = d1y(u,n)*d1y(u,n)/c^2 - snorm^2; .sort L fsum = felas(mu) + lfsing(mu); .sort * check units Hide felas; Hide fsum; L lfsingunits = lfsing(mu); L felasunits = felas(mu); L fsumunits = fsum; Id c^p? = (L/T)^p; Id a(mu) = L/T^2; Id u(mu) = L/T; Id eta(mu,i?) = 1; Id d1y(u,i?) = L/T; Id d1u(mu?,i?) = 1/T; Id d1y(a,i?) = L/T^2; Id d1y(mu?,i?) = 1; Id d2y(u,i1?,i2?) = 1/T; Id d2y(mu?,i1?,i2?) = 1/L; Id eps^p? = L^p; Id snorm^p? = 1; #if 0 * effective only if U not substituted (debugging) Id [d2U/dalp.dzet](i?) = U/L; Id [d2U/dzet.dalp](i?) = U/L; Id [dU/dzet](i1?,i2?) = U; #endif .sort Unhide felas; Unhide fsum; G flor(mu) = lfsing(mu); Id d1y(mu?,n) = eta(mu,n) + u(mu)*d1y(u,n)/c^2; Id u.u = c^2; Id u.a = 0; Id eta(u,n?)= 0; Id d1u(u,n) = 0; B eps,snorm; *B eps,snorm,u,a,eta,d1y,d1u,d2y; Print +s; .store; felas(mu) = + eps^-2*snorm^-3 * ( - 1/2*eta(mu,n) ) + eps^-1*snorm^-5 * ( + 3/2*d1y(u,n)*d1y(u,n)*eta(a,n)*eta(mu,n)*c^-4 - 3*d1y(u,n)*d1u(N1_?,n)*eta(mu,n)*eta(N1_?,n)*c^-2 + 3/2*d2y(N1_?,n,n)*eta(mu,n)*eta(N1_?,n) ) + eps^-1*snorm^-3 * ( + 1/2*d1y(u,n)*d1y(u,n)*a(mu)*c^-4 - d1y(u,n)*d1u(mu,n)*c^-2 - 1/2*d2y(u,n,n)*u(mu)*c^-2 + 1/2*d2y(mu,n,n) - 1/2*eta(a,n)*eta(mu,n)*c^-2 ) + eps^-1*snorm^-1 * ( + 1/2*a(mu)*c^-2 ); fsum = + eps^-2*snorm^-3 * ( - 3/2*eta(mu,n) ); lfsingunits = + L^-2 ; felasunits = - L^-2 ; fsumunits = 0; flor(mu) = + eps^-2*snorm^-3 * ( - eta(mu,n) ) + eps^-1*snorm^-5 * ( - 3/2*d1y(u,n)*d1y(u,n)*eta(a,n)*eta(mu,n)*c^-4 + 3*d1y(u,n)*d1u(N1_?,n)*eta(mu,n)*eta(N1_?,n)*c^-2 - 3/2*d2y(N1_?,n,n)*eta(mu,n)*eta(N1_?,n) ) + eps^-1*snorm^-3 * ( - 1/2*d1y(u,n)*d1y(u,n)*a(mu)*c^-4 + d1y(u,n)*d1u(mu,n)*c^-2 + 1/2*d2y(u,n,n)*u(mu)*c^-2 - 1/2*d2y(mu,n,n) + 1/2*eta(a,n)*eta(mu,n)*c^-2 ) + eps^-1*snorm^-1 * ( - 1/2*a(mu)*c^-2 ); Save elasemPE.sav felas; .end 0.00 sec out of 0.00 sec