Action-minimizing periodic orbits of the Lorentz force equation with dominant vector potential

TL;DR

This paper proves the existence of non-constant periodic solutions to the Lorentz force equation driven primarily by the vector potential, using a variational approach that handles singular potentials and sign-indefinite functionals.

Contribution

It introduces a novel variational method to establish periodic orbits in the Lorentz force equation emphasizing the vector potential's role, even with singular scalar potentials.

Findings

Existence of non-constant periodic solutions without scalar potential

Extension to cases with singular scalar potentials

Novel variational techniques exploiting sign-indefiniteness

Abstract

We establish the existence of non-constant periodic solutions to the Lorentz force equation, where no scalar potential is needed to induce the electromagnetic field. Our results extend to cases where a possibly singular scalar potential is present, although the vector potential assumes a leading role. The approach is based on minimizing the action functional associated with the relativistic Lagrangian. The compactness of the minimizing sequences requires the existence of negative values for the functional, which is proven using novel ideas that exploit the sign-indefinite nature of the term involving the vector potential.