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

TL;DR

The paper proves that the $S^1$-invariance of the Poincaré action functional for the Lorentz force equation yields multiple periodic solutions with a fixed period. It develops an abstract Lusternik–Schnirelman framework for nonsmooth functionals using the $S^1$-index and a weak compactness setting, and employs Ekeland–Lasry regularization and Ghoussoub localization to obtain multiplicity results. Applying the theory to autonomous electric and magnetic potentials, it shows that for large coupling $\lambda$ there are at least $3m$ distinct $2\pi$-periodic orbits, i.e., $T$-periodic solutions, with $T=2\pi$, by locating negative-level critical orbits of the Poincaré functional. The work generalizes classical index results and provides concrete conditions under which the number of periodic Lorentz-force trajectories grows with the interaction strength, including a representative example $V(q)=\arctan(\lambda|q|^2)$.

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.

Theorems & Definitions (24)

• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Theorem 1
• Lemma 6
• Theorem 2
• Remark 1
• Theorem 3