  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.
