On the Riemannian description of chaotic instability in Hamiltonian dynamics

In this work we investigate Hamiltonian chaos using elementary Riemannian geometry. This is possible because the trajectories of a standard Hamiltonian system (i.e., having a quadratic kinetic energy term) can be seen as geodesics of the configuration space manifold equipped with the standard Jacobi metric. The stability of the dynamics is tackled with the Jacobi-Levi-Civita equation (JLCE) for geodesic spread and is applied to the case of a two degrees of freedom Hamiltonian. A detailed comparison is made among the qualitative informations given by Poincare sections and the results of the geometric investigation. Complete agreement is found. The solutions of the JLCE are also in quantitative agreement with the solutions of the tangent dynamics equation. The configuration space manifold associated to the Hamiltonian studied here is everywhere of positive curvature. However, curvature is not constant and its fluctuations along the geodesics can yield parametric instability of the trajectories, thus chaos. This mechanism seems to be one of the most effective sources of chaotic instabilities in Hamiltonians of physical interest, and makes a major difference with Anosov flows, and, in general, with abstract geodesic flows of ergodic theory.