Periodic Orbits of Twisted Geodesic Flows and The Weinstein-Moser Theorem

TL;DR

This paper proves the existence of periodic orbits in twisted geodesic flows under certain conditions, extending the Weinstein-Moser theorem using Floer homology and Sturm theory.

Contribution

It generalizes the Weinstein-Moser theorem to broader settings involving symplectic extrema and twisted geodesic flows.

Findings

Periodic orbits exist on low energy levels for twisted geodesic flows with symplectic, spherically rational magnetic fields.

All energy levels near Morse-Bott non-degenerate extrema carry periodic orbits under certain topological conditions.

The proof combines Sturm theory and Floer homology techniques.

Abstract

In this paper, we establish the existence of periodic orbits of a twisted geodesic flow on all low energy levels and in all dimensions whenever the magnetic field form is symplectic and spherically rational. This is a consequence of a more general theorem concerning periodic orbits of autonomous Hamiltonian flows near Morse-Bott non-degenerate, symplectic extrema. Namely, we show that all energy levels near such extrema carry periodic orbits, provided that the ambient manifold meets certain topological requirements. This result is a partial generalization of the Weinstein-Moser theorem. The proof of the generalized Weinstein-Moser theorem is a combination of a Sturm-theoretic argument and a Floer homology calculation.