  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.
