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.