title: Solutions of the Wheeler-Feynman equations with discontinuous velocities.
abstract: We generalize Wheeler-Feynman electrodynamics with a variational boundary value problem for continuous boundary segments that might include velocity discontinuity points. Critical-point orbits must satisfy the Euler-Lagrange equations of the action functional at most points, which are neutral differential delay equations (the Wheeler-Feynman equations of motion). At velocity discontinuity points, critical-point orbits must satisfy the Weierstrass-Erdmann continuity conditions for the partial momenta and the partial energies. We study a special setup having the shortest time-separation between the (infinite-dimensional) boundary segments, for which case the critical-point orbit can be found using a two-point boundary problem for an ordinary differential equation. For this simplest setup, we prove that orbits can have discontinuous velocities. We construct a numerical method to solve the Wheeler-Feynman equations together with the Weierstrass-Erdmann conditions and calculate some numerical orbits with discontinuous velocities. We also prove that the variational boundary value problem has a unique solution depending continuously on boundary data, if the continuous boundary segments have velocity discontinuities along a reduced local space.
authors: Departamento de Física Matemática, Instituto de Física, Universidade de São Paulo, Rua do Matão, Travessa R, Nr. 187, Caixa Postal 66318, CEP 05508-090 São Paulo, São Paulo, Brasil.
authors: Daniel Câmara
authors: DC
authors: de Souza
authors: Departamento de Física, Universidade Federal de São Carlos, Rodovia Washington Luis, km 235 Caixa Postal 676, CEP 13565-905 São Carlos, São Paulo, Brazil.
authors: Jayme
authors: J
authors: De Luca