Regularization of the collision in the electromagnetic two-body problem

Abstract

We derive a differential equation that is regular at the collision of two equal-mass bodies with attractive interaction in the relativistic action-at-a-distance electrodynamics. Our method uses the energy constant related to the Poincaré invariance of the theory to motivate the regularizing coordinate transformation and to remove infinities from the equation of motion. The collision orbits are calculated numerically using the regular equation adapted in a self-consistent minimization method (a stable numerical method that chooses only nonrunaway solutions). This dynamical system appeared 100 years ago as a time-symmetric relativistic motion and aquired the status of electrodynamics in the 1940's by the works of Dirac, Wheeler and Feynman. We outline the method with an emphasis on the physics of this complex conservative dynamical system.