  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.
