Non-contractible periodic orbits of Hamiltonian flows on twisted cotangent bundles

For many classes of symplectic manifolds, the Hamiltonian flow of a function with sufficiently large variation must have a fast periodic orbit. This principle is the base of the notion of Hofer-Zehnder capacity and some other symplectic invariants and leads to numerous results concerning existence of periodic orbits of Hamiltonian flows. Along these lines, we show that given a negatively curved manifold M, a neigbhourhood U of M in the cotangent bundle, a sufficiently small magnetic field and a non-trivial free homotopy class of loops, then the magnetic flow of a Hamiltonian with big enough variation has a one-periodic orbit in that class. As a consequence, we obtain estimates for the relative Hofer-Zehnder capacity and the Biran-Polterovich-Salamon capacity of a neighbourhood of M.