On the existence of closed magnetic geodesics via symplectic reduction

TL;DR

This paper uses symplectic reduction to interpret the existence of closed magnetic geodesics as a critical point problem, linking stability properties of energy hypersurfaces and coisotropic submanifolds in symplectic geometry.

Contribution

It introduces a novel symplectic reduction framework to study magnetic geodesics as critical points of action functionals, connecting stability analysis with geometric structures.

Findings

Reformulation of magnetic geodesic problem as a critical point problem

Establishment of relations between stability of energy hypersurfaces and coisotropic submanifolds

Reproof of a key result in magnetic geodesics using symplectic reduction

Abstract

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 $(T^*M,dp\wedge dq+\pi^*\sigma)$ and of the corresponding codimension 2 coisotropic submanifolds in $(T^*E,dp\wedge dq)$ arising via symplectic reduction. Finally, we reprove the main result of [9] in this setting.