On an Integrable Magnetic Geodesic Flow on the Two-torus

The magnetic geodesic flow on a flat two-torus with the magnetic field $F=\cos(x)dx\wedge dy$ is completely integrated and the description of all contractible periodic magnetic geodesics is given. It is shown that there are no such geodesics for energy $E\geqslant1/2$, 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.