Equations of Motion for Variational Electrodynamics

Abstract

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.

21 pages, 1 figure, published in Journal of Differential Equations, 260 (2016) 5816-5833