On an integrable magnetic geodesic flow on the two-torus

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 $E\ge 1/2$, 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.

arXiv comment18 pages

References in corpus (2)

• On the existence of periodic orbits for magnetic systems on the two-sphere
• Periodic magnetic geodesics on almost every energy level via variational methods