Periodic magnetic geodesics on almost every energy level via variational methods

TL;DR

This paper demonstrates that for almost every energy level, periodic magnetic geodesics can be found using variational methods, even when the action functional is unbounded from below, by employing the principle of throwing out cycles.

Contribution

It introduces a novel application of the principle of throwing out cycles to establish the existence of periodic magnetic geodesics on almost all energy levels.

Findings

Periodic magnetic geodesics exist on almost every energy level.

The principle of throwing out cycles effectively finds critical points despite unbounded action.

The method applies to strong exact magnetic fields.

Abstract

For strong exact magnetic fields the action functional (i.e., the length plus the linear magnetic term) is not bounded from below on the space of closed contractible curves and the lower estimates for critical levels are derived by using the principle of throwing out cycles. It is proved that for almost every energy level the principle of throwing out cycles gives periodic magnetic geodesics on the critical levels defined by the "thrown out" cycles.