Infinitely many periodic solutions to a Lorentz force equation with singular electromagnetic potential

TL;DR

This paper proves the existence of infinitely many periodic solutions to a relativistic Lorentz force equation with singular electromagnetic potentials, using non-smooth critical point theory and min-max principles.

Contribution

It introduces a novel application of Lusternik-Schnirelmann type min-max methods to find multiple periodic solutions in a relativistic setting with singular fields.

Findings

Proved existence of infinitely many T-periodic solutions.

Applied results to charged particle motion under Liénard-Wiechert potentials.

Extended methods to the relativistic forced Kepler problem.

Abstract

We consider the Lorentz force equation $$ \frac{d}{dt}\left(\frac{m\dot{x}}{\sqrt{1-|\dot{x}|^{2}/c^{2}}}\right) = q \left(E(t,x) + \dot x \times B(t,x)\right), \qquad x \in \mathbb{R}^3, $$ in the physically relevant case of a singular electric field $E$. Assuming that $E$ and $B$ are $T$-periodic in time and satisfy suitable further conditions, we prove the existence of infinitely many $T$-periodic solutions. The proof is based on a min-max principle of Lusternik-Schrelmann type, in the framework of non-smooth critical point theory. Applications are given to the problem of the motion of a charged particle under the action of a Li\'enard-Wiechert potential and to the relativistic forced Kepler problem.