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

Abstract

In this paper we prove that the $S^1$-invariance of the Poincaré 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 Poincaré functional.

25 pages; to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci