On the existence of closed magnetic geodesics via symplectic reduction

Let (M, g) be a closed Riemannian manifold and sigma be a closed 2-form on M representing an integer cohomology class. In this paper, using symplectic reduction, we show how the problem of existence of closed magnetic geodesics for the magnetic flow of the pair (g, sigma) can be interpreted as a critical point problem for a Rabinowitz-type action functional defined on the cotangent bundle T* E of a suitable S-1-bundle E over M or, equivalently, as a critical point problem for a Lagrangian-type action functional defined on the free loopspace of E. We then study the relation between the stability property of energy hypersurfaces in (T* M, dp boolean AND dq + pi* sigma) and of the corresponding codimension 2 coisotropic submanifolds in (T* E, dp boolean AND dq) arising via symplectic reduction. Finally, we reprove the main result of Asselle and Benedetti (J Topol Anal 8(3):545-570, 2016) in this setting.