  We generalize Wheeler-Feynman electrodynamics by the minimization of a finite
action functional defined for variational trajectories that are required to
merge continuously into given past and future boundary segments. We prove that
the boundary-value problem is well-posed for two classes of boundary data and
show that the well-posed solution in general has velocity discontinuities,
henceforth broken extrema. Along regular segments, broken extrema satisfy the
Euler-Lagrange neutral differential delay equations with state-dependent
deviating arguments. At points where velocities are discontinuous, broken
extrema satisfy the Weierstrass-Erdmann conditions that energies and momenta
are continuous. The electromagnetic fields of the variational trajectories are
derived quantities that can be extended only to a bounded region B of
space-time. For extrema with a finite number of velocity discontinuities,
extended fields are defined for all point in B with the exception of sets of
zero measure. The extended fields satisfy the integral laws of classical
electrodynamics for most surfaces and curves inside B. As an application, we
study globally bounded trajectories with vanishing far-fields for the
hydrogenoid atomic models of hydrogen, muonium and positronium. Our model uses
solutions of the neutral differential delay equations along regular segments
and a variational approximation for the collisional segments. Each hydrogenoid
model predicts a discrete set of finitely measured neighbourhoods of orbits
with vanishing far-fields at the correct atomic magnitude and in quantitative
and qualitative agreement with experiment and quantum mechanics, i.e., the
spacings between consecutive discrete angular momenta agree with Planck's
constant within thirty-percent, while orbital frequencies agree with a
corresponding spectroscopic line within a few percent.
