The geometry of reaction dynamics

Abstract

The geometrical structures which regulate transformations in dynamical systems with three or more degrees of freedom form the subject of this paper. Our treatment focusses on the 2n 3 dimensional normally hyperbolic invariant manifold (NHIM) in n-degree-of-freedom systems associated with a center center saddle in the phase space ow in the 2n 1 dimensional energy surface. The NHIM bounds a 2n 2 dimensional surface, called a \\transition state" in chemical reaction dynamics, which partitions the energy surface into volumes characterized as \\before" and \\after" the transformation. This surface is the long-sought momentum-dependent transition state beyond two degrees of freedom. The 2n 2 dimensional stable and unstable manifolds associated with the 2n 3 dimensional NHIM are impenetrable barriers with the topology of multidimensional spherical cylinders. The phase ow interior to these spherical cylinders passes through the transition state as the system undergoes its transformation. The phase ow exterior to these spherical cylinders is directed away from the transition state and, consequently, will never undergo the transition. The explicit forms of these phase space barriers can be evaluated using normal form theory. Our treatment has the advantage of supplying a practical algorithm, and we demonstrate its use on a strongly coupled nonlinear Hamiltonian, the hydrogen atom in crossed electric and magnetic elds. 1 1