Riemannian theory of Hamiltonian chaos and Lyapunov exponents

It has been known for a long time that the solutions of Newton equations can be seen as geodesics of suitably defined Riemannian manifolds. After the early attempts of N.S. Krylov, dating back to the ‘40s of the last century, further efforts to resort to the mentioned Riemannian framework to explain the origin of chaos in Newtonian/Hamiltonian dynamical systems invariably failed. The a-priori assumption that chaos would only stem from hyperbolicity of the mechanical manifolds seems to be the reason for these failures. Actually, numerical experiments have unveiled another mechanism typically responsible for the instability of geodesic flows associated with physically meaningful Hamiltonians. Hence, under some simplifying hypotheses, the Jacobi – Levi-Civita equation for geodesic spread can be approximated by an effective equation which formally describes a stochastic oscillator; an analytic formula for the instability growth rate of its solutions is worked out providing an analytic way of estimating the largest Lyapunov exponent. Applied to the Fermi-Pasta-Ulam beta-model and to a one-dimensional XY model this analytic formula provides a strikingly good agreement with numeric values of the largest Lyapunov exponent computed by means of the standard tangent dynamics equation.