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

Abstract

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.

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

“…(ii) Let furthermore T 0 < −1 be a sufficiently large negative number such that x(t) > y(t) and |ẋ(t)|, |ẏ(t ′ )| < 1 for all t ≤ T 0 and t ′ ≤ T + 0 is fulfilled, and let (a T , b T ) T ≤T 0 denote the family of conditional solutions to the FST equations (1) inferred by Theorem 1 satisfying the initial conditions (4), (5). Then, there is a sequence (T n ) n∈N with lim n→∞ T n = −∞ fulfilling lim…”

“…Before we discuss the proof, we remark that (27) is a corollary to Proposition 2.1 from [5], but (28) is deduced from Lemma 1. This poses no problem since (28) will only be exploited in the proof of the main result Theorem 2, whereas the estimate (27) will be used throughout the proof of Lemma 2.…”

“…In case the advanced terms are omitted, the respective equations are called Synge equations. For this case, solutions on the half-line can be found by integration, but obtaining global ones is still highly non-trivial; see [7,1,5]. The goal of this work, i.e., Theorem 2 below, is to ensure global existence of solutions to equations (1) including both the advanced and retarded terms.…”

“…In a recent work [5], for two charges on the straight line, it was possible to infer from prescribed initial data existence of solutions not only in a finite time interval but on the half-line. More precisely, it was shown in [5, Theorem 1.1]:…”

“…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 [5], where existence of FST solutions was proven on the half-line, to ensure global existence -a result that had been obtained by Bauer [2] 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.…”