  We study the dynamics and symplectic topology of energy hypersurfaces of
mechanical Hamiltonians on twisted cotangent bundles. We pay particular
attention to periodic orbits, displaceability, stability and the contact type
property, and the changes that occur at the Mane critical value c. Our main
tool is Rabinowitz Floer homology. We show that it is defined for hypersurfaces
that are either stable tame or virtually contact, and it is invariant under
under homotopies in these classes. If the configuration space admits a metric
of negative curvature, then Rabinowitz Floer homology does not vanish for
energy levels k>c and, as a consequence, these level sets are not displaceable.
We provide a large class of examples in which Rabinowitz Floer homology is
non-zero for energy levels k>c but vanishes for k<c, so levels above and below
c cannot be connected by a stable tame homotopy. Moreover, we show that for
strictly 1/4-pinched negative curvature and non-exact magnetic fields all
sufficiently high energy levels are non-stable, provided that the dimension of
the base manifold is even and different from two.
