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

TL;DR

This paper studies the mathematical properties of electrodynamic two-body problems with delays, proving existence and uniqueness of solutions for various models describing point charges on a line.

Contribution

It provides the first rigorous existence results for Fokker-Schwarzschild-Tetrode and Synge equations in a straight-line two-charge setup, including criteria for solution uniqueness.

Findings

Existence of solutions for FST equations on half-lines.

Global solutions for Synge equations with Newtonian data.

A criterion for unique solution determination via finite trajectory data.

Abstract

Due to the finite speed of light, direct electrodynamic interaction between point charges can naturally be described by a system of ordinary differential equations involving delays. As electrodynamics is time-symmetric, these delays appear as time-like retarded as well as advanced arguments in the fundamental equations 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. Dropping radiation terms, which in many situations are very small, the latter equations are called Synge equations. In both cases, few mathematical results are available on existence of solutions, and even fewer on uniqueness. 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 strips; 2) Global existence of the Synge equations for Newtonian Cauchy data; 3) Global existence of a FST toy model that involves advanced and retarded terms. Furthermore, we give a sufficient criterion that uniquely distinguishes solutions by means of finite trajectory strips.