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

Abstract

We consider the Lorentz force equationddt(mx˙1−|x˙|2/c2)=q(E(t,x)+x˙×B(t,x)),x∈R3, 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-Schnirelmann 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.