Electrodynamic two-body problem for prescribed initial data on a straight line

Abstract

Electrodynamic interaction between point charges can be described by a system of ODEs involving advanced and retarded delays – the so-called Fokker–Schwarzschild–Tetrode (FST) equations. In special situations, approximate equations can be derived which are purely retarded. Upon omission of the terms describing radiation friction, these are called Synge equations. In both cases, few mathematical results are available on existence and uniqueness of solutions. We investigate the situation of two like point-charges in 3+1 space–time dimensions restricted to motion on a straight line. We give a priori estimates on the asymptotic motion and, using a Leray–Schauder argument, prove: 1) Existence of solutions to the FST equations on the future or past half-line given finite trajectory segments; 2) Global existence of the Synge equations for Cauchy data; 3) Global existence of a FST toy model. Furthermore, we give a sufficient criterion that uniquely distinguishes solutions by means of finite trajectory segments.