Mountain pass periodic solutions for the Lorentz force equation via the Poincaré action functional

Abstract

In this paper, we study the Lorentz force equation with the rest mass [Formula: see text] and periodic boundary conditions on a fixed interval [Formula: see text] [Formula: see text] where [Formula: see text] are the electric and magnetic fields. From the Poincaré paper concerning the special relativity it is well known that this is the Euler–Lagrange equation of the action functional given by [Formula: see text] defined for all [Formula: see text]-periodic Lipschitz functions [Formula: see text] such that [Formula: see text] In this paper, under some assumptions on the potentials [Formula: see text] and [Formula: see text] around zero and infinity, we prove that [Formula: see text] has nonzero critical points which are [Formula: see text]-periodic solutions of the Lorentz force equation. To prove our main results we use new “mountain pass” methods for the Poincaré action functional [Formula: see text]