Solutions of the Two-Body Problem in Classical Action-at-a-Distance Electrodynamics: Straight-Line Motion

The equations of motion for two classical particles with equal charges and masses are solved for the case of motion confined to one dimension by the initial conditions. The charges interact via half-retarded plus half-advanced Li\'enard-Wiechert potentials. The equations, which are of differential-difference type, are solved by successive approximations and the solutions are compared with those obtained by perturbation theory. The results suggest the existence of a minimum distance of closest approach.