Geometrical hints for a nonperturbative approach to Hamiltonian dynamics

Abstract

In this paper it is shown that elementary tools of Riemannian differential geometry can be successfully used to explain the origin of Hamiltonian chaos beyond the usual picture of homoclinic intersections. This approach stems out of first principles of mechanics and fundamental tools of Riemannian geometry. Natural motions of Hamiltonian systems can be viewed as geodesics of the configuration-space manifold M equipped with a suitable metric ${\mathit{g}}_{\mathit{J}}$, and the stability properties of such geodesics can be investigated by means of the Jacobi–Levi-Civita equation for geodesic spread. The study of the relationship between chaos and the curvature properties of the configuration-space manifold is the main concern of the present paper and is carried out by numerical simulations. Two different mechanisms for chaotic instability are found: (i) the trajectories are ‘‘scattered’’ by random encounters of regions of negative curvature (either scalar or Ricci curvature—it depends on the averaging procedure adopted); (ii) the ‘‘bumpiness’’ of (M,${\mathit{g}}_{\mathit{J}}$) yields oscillations of the Ricci curvature along the geodesics so that parametric resonance makes them unstable also in regions of positive curvature.