Numerical solutions to two-body problems in classical electrodynamics: Head-on collisions with retarded fields and radiation reaction. I. Repulsive case

Abstract

Classical trajectories of two particles with like charges have been computed numerically for head-on collisions. The trajectories are physical solutions of the Lorentz-Dirac equation with retarded fields. To eliminate runaway solutions, the third-order equation has been integrated numerically backward in time. Results are presented both for the static case (one particle is infinitely massive) and for two particles of equal mass. In the latter case, iterations are required in order to obtain self-consistent trajectories. Compared to results with the Lorentz equation, in which radiation reaction is ignored, maximum accelerations are markedly smaller, distances of closest approach are larger, and there is a small loss in particle energy rather than a large gain. No evidence was found for a lower bound on the distance of closest approach or for an upper bound on the radiated energy.