On The Geometry of Hamiltonian Chaos

We show that Gutzwiller’s characterization of chaotic Hamiltonian systems in terms of the curvature associated with a Riemannian metric tensor in the structure of the Hamiltonian can be extended to a wide class of potential models of standard form through definition of a conformal metric. The geodesic equations reproduce the Hamilton equations of the original potential model when a transition is made to the dual manifold, and the geodesics in the dual space coincide with the orbits of the Hamiltonian potential model. We therefore find a direct geometrical description of the time development of a Hamiltonian potential model. The second covariant derivative of the geodesic deviation in this dual manifold generates a dynamical curvature, resulting in (energy dependent) criteria for unstable behavior different from the usual Lyapunov criteria. We discuss some examples of unstable Hamiltonian systems in two dimensions giving, in particular, detailed results for a potential obtained from a fifth order expansion of a Toda lattice Hamiltonian. PACS: 45.20.Jj, 47.10.Df, 05.45.-a, 05.45.Gg Gutzwiller has pointed out that a Hamiltonian system of the form (we use the summation convention) H = 1 2M gijp p , (1) where gij is a function of the coordinates alone, can have chaotic orbits if the curvature associated with the metric gij is negative. The resulting curvature is a property of the manifold of solutions for the equations of motion generated by this Hamiltonian. One can easily see that the orbits described by the Hamilton equations for (1) coincide with the geodesics on a Riemannian space associated with the metric gij , 1,2 i.e., it follows directly from the Hamilton equations associated with (1) that (using the time derivative of (11) and (12) below) ẍl = −Γ mn l ẋmẋn, (2) where the connection form is given by Γ l = 1 2 glk {∂g ∂xn + ∂g ∂xm − ∂g ∂xk }