Analytic computation of the strong stochasticity threshold in Hamiltonian dynamics using Riemannian geometry

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Abstract
This paper concerns the study of the strong stochasticity threshold (SST) in Hamiltonian systems with many degrees of freedom, and more specifically the stability problem of this threshold in the thermodynamic limit (N→∞). The investigation is based on a recently proposed differential geometrical description of Hamiltonian chaos. The mathematical framework is given by Eisenhart’s formulation of Newtonian mechanics in a suitably enlarged configuration space-time: a Riemannian manifold equipped with an affine metric. Using the Jacobi–Levi-Civita equation for geodesic spread, we establish a relation between curvature properties of the ambient manifold and the stability—or instability—of the dynamics. The use of the Eisenhart metric makes it clearly evident that a dominating source of chaoticity in Hamiltonian flows of phyical interest is represented by parametric resonance induced by curvature fluctuations along the geodesics and not by negativeness of some curvature property. Here only Ricci curvature is involved because the scalar curvature vanishes identically with this metric. Thus a geometric quantity relevant to the study of chaos is the degree of bumpiness of the ambient manifold, i.e., the integral of the Ricci curvature carried over the whole manifold with the constraint of energy constancy. This quantity, considered as a function of the energy density ɛ, clearly marks the SST. A simple sufficient criterion is given to identify integrable systems and is here applied to a chain of linear oscillators as well as to the Toda lattice. In the case of the Fermi-Pasta-Ulam β model we have worked out analytically the ɛ dependence of the degree of bumpiness of the ambient manifold. This allowed us to prove that the SST is stable in the N→∞ limit. An operational definition of the critical energy density is also provided and shown to yield predictions in excellent agreement with previous results based on Lyapunov exponents.