Geometric description of chaos in two-degrees-of-freedom Hamiltonian systems.

Abstract

This paper tackles Hamiltonian chaos by means of elementary tools of Riemannian geometry. The stability of dynamics, related to curvature properties of the configuration space manifold, is investigated through the Jacobi--Levi-Civita equation (JLC) for geodesic spread. The case of two-degrees-of-freedom Hamiltonians is considered in general and is applied to the H\'enon-Heiles model. The detailed qualitative information provided by Poincar\'e sections are compared with the results of geometric investigation; a complete agreement is found. The solutions of the JLC equation are also in quantitative agreement with the solutions of the tangent dynamics equation. It is shown here that chaos in the H\'enon-Heiles model stems from parametric instability due to positive curvature fluctuations along the geodesics (dynamical motions) of configuration space manifold. This mechanism is apparently the most relevant---and in many cases unique---source of chaoticity in physically meaningful Hamiltonians. Hence a major difference with the geometric description of chaos in abstract ergodic theory is found; chaotic Hamiltonian flows of physics have nothing to do with Anosov flows defined on negative curvature manifolds. Even in the case of fully developed Hamiltonian chaos, hyperbolicity is not necessarily involved. Finally, the paper deals with the problem of finding other criteria for the onset of chaos based on purely geometric tools and independently of the numerical knowledge of the trajectories. \textcopyright{} 1996 The American Physical Society.