Equations of motion for variational electrodynamics

We extend the variational problem of Wheeler–Feynman electrodynamics by generalizing the electromagnetic functional to a local space of absolutely continuous trajectories possessing a derivative (velocities) of bounded variation. We show here that the Gateaux derivative of the generalized functional defines two partial Lagrangians for variations in our generalized local space, one for each particle. We prove that the critical-point conditions of the generalized variational problem are: (i) the Euler–Lagrange equations must hold Lebesgue-almost-everywhere and (ii) the momentum of each partial Lagrangian and the Legendre transform of each partial Lagrangian must be absolutely continuous functions, generalizing the Weierstrass–Erdmann conditions.