Free oscillations of prescribed energy at a saddle-point of the potential in Hamiltonian dynamics

We consider a class of Hamiltonians of the form <p,p>+V(q), where the potential is even and has (a possibly highly degenerate) critical point which is a saddle at 0∈ℝ n and V(0)=0. Under certain natural conditions on V we show that the corresponding Hamiltonian system has non-trivial periodic orbits of all positive energies passing through 0. The methods employed give an illustration of the significance of monotone trajectories in the study of certain classical Hamiltonian systems.