Electrodynamic Two-Body Problem for Prescribed Initial Data on the Straight Line

摘要

Due to the finite speed of light, direct electrodynamic interaction betweenpoint charges can naturally be described by a system of ordinary differentialequations involving delays. As electrodynamics is time-symmetric, these delaysappear as time-like retarded as well as advanced arguments in the fundamentalequations of motion -- the so-called Fokker-Schwarzschild-Tetrode (FST)equations. However, for special initial conditions breaking the time-symmetry,effective equations can be derived which are purely retarded. Droppingradiation terms, which in many situations are very small, the latter equationsare called Synge equations. In both cases, few mathematical results areavailable on existence of solutions, and even fewer on uniqueness. Weinvestigate the situation of two like point-charges in $3+1$ space-timedimensions restricted to motion on a straight line. We give a priori estimateson the asymptotic motion and, using a Leray-Schauder argument, prove: 1)Existence of solutions to the FST equations on the future or past half-linegiven finite trajectory strips; 2) Global existence of the Synge equations forNewtonian Cauchy data; 3) Global existence of a FST toy model that involvesadvanced and retarded terms. Furthermore, we give a sufficient criterion thatuniquely distinguishes solutions by means of finite trajectory strips.