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

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

2007-05-13

arxiv.org/abs/0705.1818v1

Mathematics

Symplectic Geometry

34 pages, 2 figures

Scientific paper

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.

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