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.The geometric approach is intrinsically nonperturbative because everything is well defined at any energy, i.e., quasi-integrability is not required as in the case of classical perturbation theory. Therefore this approach is fit to describe the existence of the strong-stochasticity threshold (SST) in high-dimensional Hamiltonian flows. This threshold refers to a transition between weak and strong chaoticity of the dynamics and, correspondingly, between slow and fast mixing in phase space. In view of applications to equilibrium and nonequilibrium statistical mechanics, the SST appears as the transition feature of high dimensional Hamiltonian flows with the greatest physical significance. As the SST concerns chaotic dynamical behaviors, its existence cannot be understood within the framework of classical perturbation theory, whereas it is shown that the SST can be related to some major geometrical change of the constant-energy surfaces of phase space. A clear-cut distinction can be made between integrable and nonintegrable systems. Finally, a new meaning is given to the standard algorithm to compute numerical Lyapunov exponents, and it is shown that Oseledets multiplicative theorem is not necessary to justify it.