Neutral Delay and a Generalization of Electrodynamics

The equations for the electromagnetic two-body problem are neutral-delay equations that for generic initial data have solutions with discontinuous derivatives. If one wants to use these neutral-delay equations with arbitrary initial data, solutions with discontinuous derivatives must be allowed. Surprisingly, this same neutrality is compatible with the recently developed variational method with mixed-type boundaries for the Wheeler-Feynman electrodynamics. We show that two-body electromagnetic orbits with discontinuous velocities are physically necessary by showing that orbits with vanishing far-fields amost everywhere must have some discontinuous velocities on a few points. We generalize the Wheeler-Feynman electrodynamics with the variational method to include all continuous trajectories, allowing piecewise-differentiable weak solutions represented by trajectories with fields defined almost everywhere (but on a set of points of zero measure where velocities jump). Along with this generalization we formulate the generalized absorber hypothesis that the far-fields vanish asymptotically almost everywhere and show that bounded two-body orbits satisfying the generalized absorber hypothesis need to have discontinuous derivatives on a few points. We also give the general solution for the family of bounded non-radiating two-body orbits. We discuss the physics of orbits with discontinuous derivatives and show that these conserve the physical momentum, stressing the differences to classical variational methods. Last, we discuss how the electromagnetic variational method with mixed-type boundaries is well-posed but lacks reversibilty.