Global solutions to the electrodynamic two-body problem on a straight line

Abstract

The classical electrodynamic two-body problem has been a long standing open problem in mathematics. For motion constrained to the straight line, the interaction is similar to that of the two-body problem of classical gravitation. The additional complication is the presence of unbounded state-dependent delays in the Coulomb forces due to the finiteness of the speed of light. This circumstance renders the notion of local solutions meaningless, and therefore, straight-forward ODE techniques can not be applied. Here, we study the time-symmetric case, i.e., the Fokker-Schwarzschild-Tetrode (FST) equations, comprising both advanced and retarded delays. We extend the technique developed in \cite{DirkGuenter}, where existence of FST solutions was proven on the half-line, to ensure global existence -- a result that had been obtained by Bauer \cite{Bauer} in 1997. Due to the novel technique, the presented proof is shorter and more transparent but also relies on the idea to employ asymptotic data to characterize solutions.