A two-body problem of classical electrodynamics: the one-dimensional case☆

Abstract The equations of motion for the two-body problem of classical electrodynamics can be formulated by substituting the expressions for the field of a moving charge, calculated from the Lienard-Wiechert potentials, into the Lorentz-Abraham force law. In this paper, radiation reaction is omitted and the charges are assumed to move along the x-axis. Due to the finite speed of propagation, c, of electrical effects, the differential equations involve time delays, which depend upon the unknown trajectories. Thus the equations of motion are not ordinary differential equations, and one does not determine a unique solution for t > t0 by merely specifying the positions and velocities of the charges at t0. One specifies rather arbitrary initial trajectories of the two charges over some appropriate interval [α, t0], where α t0 unless and until the charges collide. Moreover the solution depends continuously on the given initial trajectories. It is also shown that two point charges of like sign cannot collide, while two point charges of opposite signs may or may not collide, depending upon the initial data. Moreover, in the event of a collision, the velocities of the two charges become +c and −c at the instant of collision. The effect of an external electric field in the x-direction is also considered.