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

Abstract

We consider the Lorentz force equation $$\frac{d}{dt}\left(\frac{m\dot{x}}{\sqrt{1-|\dot{x}{|}^2/c^2}}\right)=q\left(E(t,x)+\dot{x}\times B(t,x)\right),x\in {\mathbb{R}}^3,$$ 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 Liénard-Wiechert potential and to the relativistic forced Kepler problem.