Numerical Solutions to Two-Body Problems in Classical Electrodynamics: Straight-Line Motion with Retarded Fields and No Radiation Reaction

Abstract

Trajectories of two charged particles of equal mass have been calculated for straight-line motion using retarded fields without radiation reaction. Both attractive and repulsive interactions are considered. For the attractive case, in which particles are released from rest with various initial separations $d$, the acceleration is shown to approach - $\sqrt{2}$ (in natural units) as the particles approach each other, and oscillations with are significant for $d\ensuremath{\lesssim}1$ are strongly damped for $d>>1$. For the repulsive case, in which the particles are thrown directly at one another with high initial velocity, no lower limit is found for the distance of closest approach as initial velocities are increased. Furthermore, the particle energy is found to increase during collision. We explain this result in terms of an energy-conservation theorem and discuss its significance with regard to the omission of radiation reaction.