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 Bott-nondegenerate 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. 
Viktor L. Ginzburg (UC Santa Cruz), Ely Kerman (Fields Institute)