Periodic orbits of Hamiltonian flows near symplectic extrema

For Hamiltonian flows we establish the existence of periodic orbits on a sequence of level sets approaching a Bottnondegenerate symplectic extremum of the Hamiltonian. As a consequence, we show that a charge on a compact manifold with a nondegenerate (i.e., symplectic) magnetic field has periodic orbits on a sequence of energy levels converging to zero.