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We show that the existence of noncontractible periodic orbits for compactlysupported time-dependent Hamiltonian on the disk cotangent bundle of a Finslermanifold provided that the Hamiltonian is sufficiently large over the zerosection. We generalize the BPS capacities and earlier constructions of Weber(2006 Duke Math. J. 133, 527-568) and other authors Biran et al (2003 DukeMath. J. 119, 65-118) to the Finsler setting. We then obtain a number ofapplications including: (1) generalizing the main theorem of Xue (2017 J.Symplectic Geom. 15, 905-936) to the Lie group setting, (2) preservation ofminimal Finsler length of closed geodesics in any given free homotopy class bysymplectomorphisms, (3) existence of periodic orbits for Hamiltonian systemsseparating two Lagrangian submanifolds, (4) existence of periodic orbits forHamiltonians on noncompact domains, (5) existence of periodic orbits forLorentzian Hamiltonian in higher dimensional case, (6) partial solution to aconjecture of Kawasaki (2016 Heavy subsets and non-contractible trajectories(arXiv:1606.01964)), (7) results on squeezing/nonsqueezing theorem on toruscotangent bundles, etc.

Floer homology in the cotangent bundle of a closed Finsler manifold and noncontractible periodic orbits