  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.
