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

TL;DR

This paper proves the existence of global solutions for the classical electrodynamic two-body problem constrained to a straight line, addressing the challenges posed by unbounded delays in the Coulomb interaction.

Contribution

It extends previous methods to establish global existence of solutions for the FST equations, incorporating advanced and retarded delays, with a shorter, more transparent proof.

Findings

Proved global existence of solutions for the FST equations on the entire real line.

Extended previous half-line existence results to the full line.

Utilized asymptotic data to characterize solutions effectively.

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.