Critical Point Theory for the Lorentz Force Equation

Abstract

In this paper we study the existence and multiplicity of solutions of the Lorentz force equation $$\left(\frac{q'}{\sqrt{1-|q'|^2}}\right)'=E(t,q) + q'\times B(t,q)$$ with periodic or Dirichlet boundary conditions. In Special Relativity, this equation models the motion of a slowly accelerated electron under the influence of an electric field E and a magnetic field B. We provide a rigourous critical point theory by showing that the solutions are the critical points in the Szulkin’s sense of the corresponding Poincare non-smooth Lagrangian action. By using a novel minimax principle, we prove a variety of existence and multiplicity results. Based on the associated Planck relativistic Hamiltonian, an alternative result is proved for the periodic case by means of a minimax theorem for strongly indefinite functionals due to Felmer.