On an integrable magnetic geodesic flow on the two-torus

TL;DR

This paper completely solves the magnetic geodesic flow on a flat two-torus with a specific magnetic field, classifying all contractible periodic geodesics and analyzing their existence based on energy levels.

Contribution

It provides a complete integration of the magnetic geodesic flow on the two-torus with a specific magnetic field and characterizes all contractible periodic geodesics.

Findings

No contractible periodic geodesics for energy E ≥ 1/2.

Existence of two S^1-families of simple periodic geodesics for E < 1/2.

Action functional is positive for these geodesics, implying no negative action periodic geodesics.

Abstract

We completely integrate the magnetic geodesic flow on a flat two-torus with the magnetic field $F = \cos (x) dx \wedge dy$ and describe all contractible periodic magnetic geodesics. It is shown that there are no such geodesics for energy $E \geq 1/2$, for $E< 1/2$ simple periodic magnetic geodesics form two $S^1$-families for which the (fixed energy) action functional is positive and therefore there are no periodic magnetic geodesics for which the action functional is negative.