Geometry of Hamiltonian Chaos

The characterization of chaotic Hamiltonian systems in terms of the curvature associated with a Riemannian metric tensor in the structure of the Hamiltonian is 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 an associated manifold. We find, in this way, a direct geometrical description of the time development of a Hamiltonian potential model. The second covariant derivative of the geodesic deviation in this associated manifold results in (energy dependent) criteria for unstable behavior different from the usual Lyapunov criteria. We discuss some examples of unstable Hamiltonian systems in two dimensions.

Figure 1

(a) The dark area shows the region of negative eigenvalues for the matrix $\mathcal{V}$ for a Hamiltonian with potential (30). The light area corresponds to physically allowable motion for $E=1/6$. The region of negative eigenvalues does not penetrate the physically accessible region in this case. (b) The dark area of negative eigenvalues for the matrix $\mathcal{V}$ is seen to penetrate deeply into the light region of physically allowable motion for $E=3$.

Figure 2

(a) A Poincaré plot in the ($y$, $p_y$) plane for $E=1/6$, indicating regular motion. (b) A Poincaré plot in the ($y$, $p_y$) plane for $E=3$, indicating a strongly chaotic behavior.