$\bf{S^1}$-index theory for the Lorentz force equation

TL;DR

This paper establishes the existence of multiple periodic solutions to the Lorentz force equation by leveraging $S^1$-invariance and Lusternik-Schnirelman methods, extending previous index theories to nonsmooth functionals.

Contribution

It introduces an abstract multiplicity theorem based on the $S^1$-index, applicable to nonsmooth functionals related to the Lorentz force equation.

Findings

Multiple periodic solutions are proven to exist.

The method extends to nonsmooth functionals.

The approach adapts Lusternik-Schnirelman theory to the problem.

Abstract

In this paper we prove that the $S^1$-invariance of the Poincar\'e action functional associated to the Lorentz force equation gives the existence of multiple critical points which are periodic solutions with a fixed period. To do this, we prove an abstract multiplicity result which is based upon the Lusternik-Schnirelman method with the $S^1$-index. The corresponding result in the context of the Fadell-Rabinowitz index is proved in Ekeland and Lasry (Ann. Math., 112 (1980)). The main feature of our abstract result is that it allows us to consider nonsmooth functionals satisfying only a weak compactness condition well adapted to the Poincar\'e functional.