Geometry and dynamics of stable and unstable cylinders in Hamiltonian systems

Abstract

Stable and unstable manifolds in phase space with an S 1 × R 1 (cylindrical) geometry are shown to exist for certain two degree of freedom Hamiltonian systems. Specific attention is given to Hamiltonian systems with potential barriers, although the concepts developed are more general. The existence of these cylinders is independent of the nature of the Hamiltonian dynamics (i.e. regular or chaotic). A detailed discussion is given where we show that appropriate Poincaré sections of the cylinders yield a map structure (which we denote as “reactive islands”) that is distinct from the usual homoclinic tangle. The cylinders have the physical property that all motion across a barrier must occur through the interior of these surfaces. The cylinders thus mediate the reaction dynamics. A kinetic mechanism based upon the properties of the cylinders is developed and tested against several numerical simulations of the reaction dynamics of a model Hamiltonian system. The threshold limiting form of the standard theory of microcanonical reaction rates is derived.