Geometric Interpretation of Chaos in Two-Dimensional Hamiltonian Systems

Time-independent Hamiltonian flows are viewed as geodesic flows in a curved manifold, so that the onset of chaos hinges on properties of the curvature two-form entering into the Jacobi equation. Attention focuses on ensembles of orbit segments evolved in 2-D potentials, examining how various orbital properties correlate with the mean value and dispersion, <K> and k, of the trace K of the curvature. Unlike most analyses, which have attributed chaos to negative curvature, this work exploits the fact that geodesics can be chaotic even if K is everywhere positive, chaos arising as a parameteric instability triggered by regular variations in K along the orbit. For ensembles of fixed energy, with both regular and chaotic segments, simple patterns connect the values of <K> and k for different segments, both with each other and with the short time Lyapunov exponent X. Often, but not always, there is a near one-to- one correlation between <K> and k, a plot of these quantities approximating a simple curve. X varies smoothly along this curve, chaotic segments located furthest from the regular regions tending systematically to have the largest X's. For regular orbits, <K> and k also vary smoothly with ``distance'' from the chaotic phase space regions, as probed, e.g., by the location of the initial condition on a surface of section. Many of these observed properties can be understood qualitatively in terms of a one-dimensional Mathieu equation.