On the periodic motions of a charged particle in an oscillating magnetic field on the two-torus

TL;DR

This paper proves that for a generic small energy level, the magnetic flow on a two-torus with an oscillating magnetic field has infinitely many periodic orbits, extending previous results to non-exact oscillating cases.

Contribution

It extends the existence of infinitely many periodic orbits in magnetic flows to the case of non-exact oscillating magnetic fields on the two-torus.

Findings

For almost every small positive energy level, infinitely many periodic orbits exist.

Extends previous results from higher genus surfaces to the two-torus case.

Includes non-exact oscillating magnetic fields, broadening applicability.

Abstract

Let $(\mathbb T^2,g)$ be a Riemannian two-torus and let $\sigma$ be an oscillating $2$-form on $\mathbb T^2$. We show that for almost every small positive number $k$ the magnetic flow of the pair $(g,\sigma)$ has infinitely many periodic orbits with energy $k$. This result complements the analogous statement for closed surfaces of genus at least $2$ [Asselle and Benedetti, Calc. Var. Partial Differential Equations, 2015] and at the same time extends the main theorem in [Abbondandolo, Macarini, Mazzucchelli, and Paternain, J. Eur. Math. Soc. (JEMS), to appear] to the non-exact oscillating case.