  The main result in this paper is the $C^{\infty}$ closing lemma for a large
family of Hamiltonian flows on $4$-dimensional symplectic manifolds, which
includes classical Hamiltonian systems. First we prove the $C^{\infty}$ closing
lemma and the $C^r$ general density theorem for geodesic flows on closed
Finsler surfaces by combining a result of Asaoka-Irie with the dual lens map
technique. Then we extend our results to Hamitonian flows with certain
restriction. We also list some applications of our results in differential
geometry and contact topology.
