On the stability of Hamiltonian systems with weakly time dependent potentials

We show here that a recently developed criterion for the stability of conservative Hamiltonian systems can be extended to Hamiltonians with weak time dependence. In this method, the geodesic equations contain the Hamilton equations of the original potential model through an inverse map in the tangent space in terms of a geometric embedding. The second covariant derivative of the geodesic deviation generates a dynamical curvature, resulting in a (energy dependent) local criterion for unstable behavior different from the usual Lyapunov criterion. We show by direct simulation that our geometrical criterion predicts correctly the stability/instability of motions, sometimes contrary to indications of the local Lyapunov method.