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