On the geometry of transport in phase space I. Transport in k-degree-of-freedom Hamiltonian systems, 2≤ k

Abstract

Over the past several years the effect of geometrical structures such as KAM tori, resonance zones, and cantori on phase space transport in two-degree-of-freedom Hamiltonian systems has been studied. In three-degree-of-freedom systems and higher, these structures no longer form complete or partial barriers to transport in the sense that they or their associated stable and unstable manifolds are no longer codimension one in the level set of the Hamiltonian. In this paper we show that in the (2 k−1)-dimensional level sets of the Hamiltonian of a class of k-degree-of-freedom Hamiltonian systems the (2 k−2)-dimensional stable and unstable manifolds of normally hyperbolic invariant (2 k−3)-dimensional spheres form partial barriers to transport. Under certain conditions, we show that lobes and turnstiles can be formed from such geometrical structures in a way that is exactly analogous to the situation for two-degree-of-freedom Hamiltonian systems. Considering a foliation of the level set of the Hamiltonian into disjoint regions separated by such partial barriers, we can then give exact formulas for the transport of volumes of phase space throughout the different regions based on the dynamics of these generalized lobes and turnstiles. On the other hand, despite the fact that these partial barriers are codimension one, for k-degree-of-freedom systems, k≥3, they may not intersect in such a way so as to divide the phase space.