We show the the existence of noncontractible periodic orbits for every compactly supported time-dependent Hamiltonian on the open unit disk cotangent bundle of a Finsler manifold provided that the Hamiltonian is sufficiently large over the zero section. This result solves a conjecture of Irie and generalizes the previous results of Biran-Polterovich-Salamon and Weber etc. We then obtain a number of applications including: (1) preservation of Finsler lengths of closed geodesics by symplectomorphisms, (2) existence of periodic orbits for Hamiltonian systems separating two Lagrangian submanifolds, (3) existence of periodic orbits for Hamiltonians on noncompact domains, (4) existence of periodic orbits for Lorentzian Hamiltonian in higher dimensional case, (5) results on squeezing/nonsqueezing theorem on torus cotangent bundles. This is a joint work with Wenmin Gong.
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