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 Mañé critical value
. Our
main tool is Rabinowitz Floer homology. We show that it is defined for hypersurfaces
that are either stable tame or virtually contact, and that it is invariant 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
and,
as a consequence, these level sets are not displaceable. We provide a large class of
examples in which Rabinowitz Floer homology is nonzero for energy levels
but vanishes
for , so levels
above and below
cannot be connected by a stable tame homotopy. Moreover, we show that for strictly
–pinched
negative curvature and nonexact magnetic fields all sufficiently high energy levels are
nonstable, provided that the dimension of the base manifold is even and different
from two.