Riemannian theory of Hamiltonian chaos and Lyapunov exponents.

A non-vanishing Lyapunov exponent λ1 provides the very definition of deterministic chaos in the solutions of a dynamical system, however no theoretical mean of predicting its value exists. This paper copes with the problem of analytically computing the largest Lyapunov exponent λ1 for many degrees of freedom Hamiltonian systems as a function of ε = E/N , the energy per degree of freedom. The functional dependence λ1(ε) is of great interest because, among other reasons, it detects the existence of weakly and strongly chaotic regimes. This aim analytic computation of λ1(ε) is successfully reached within a theoretical framework that makes use of a geometrization of newtonian dynamics in the language of Riemannian differential geometry. A new point of view about the origin of chaos in these systems is obtained independently of the standard explanation based on homoclinic intersections. Dynamical instability (chaos) is here related to curvature fluctuations of the