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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Viktor L. Ginzburg, Başak Z. Gürel, Periodic orbits of twisted geodesic flows and the Weinstein–Moser theorem. Comment. Math. Helv. 84 (2009), no. 4, pp. 865–907DOI 10.4171/CMH/184