Solutions of the Wheeler-Feynman equations with discontinuous velocities

We generalize Wheeler-Feynman electrodynamics with a variational boundary-value problem with past and future boundary segments that can include velocity discontinuity points. Critical-point trajectories must satisfy the Euler-Lagrange equations of the action functional, which are neutral-differential delay equations of motion (the Wheeler-Feynman equations of motion). At velocity discontinuity points, critical-point orbits must satisfy the Weierstrass-Erdmann conditions of continuity of partial momenta and partial energies. We study a special class of boundary data having the shortest time-separation between boundary segments, for which case the Wheeler-Feynman equations reduce to a two-point boundary problem for an ordinary differential equation. For this simple case we prove that the extended variational problem has solutions with 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.