Equations of Motion for Variational Electrodynamics

We extend the variational problem of Wheeler-Feynman electrodynamics by putting the electromagnetic functional in a local space of absolutely continuous trajectories possessing a derivative (velocities) of bounded variation. Generalizing the calculus of variations for extrema with a finite number of velocity discontinuities (breaking points), we prove that the critical-point-conditions for the two-body problem in the extended local space are Euler-Lagrange equations holding Lebesgue-almost-everywhere plus the generalized Weierstrass-Erdmann conditions that (i) the partial momenta must be absolutely continuous functions and (ii) the Legendre transforms of the partial Lagrangians (i.e, the partial energies) must be absolutely continuous functions.