Approximating Multi-Dimensional Hamiltonian Flows by Billiards

Abstract

The behavior of a point particle traveling with a constant speed in a region $D\subset R^{N}$ , undergoing elastic collisions at the regions’s boundary, is known as the billiard problem. Various billiard models serve as approximation to the classical and semi-classical motion in systems with steep potentials (e.g. for studying classical molecular dynamics, cold atom’s motion in dark optical traps and microwave dynamics). Here we develop methodologies for examining the validity and accuracy of this approximation. We consider families of smooth potentials $V_\epsilon$ , that, in the limit $\epsilon\rightarrow0$ , become singular hard-wall potentials of multi-dimensional billiards. We define auxiliary billiard domains that asymptote, as $\epsilon\rightarrow0$ to the original billiards, and provide, for regular trajectories, asymptotic expansion of the smooth Hamiltonian solution in terms of these billiard approximations. The asymptotic expansion includes error estimates in the C ^r norm and an iteration scheme for improving this approximation. Applying this theory to smooth potentials that limit to the multi-dimensional close to ellipsoidal billiards, we predict when the billiard’s separatrix splitting (which appears, for example, in the nearly flat and nearly oblate ellipsoids) persists for various types of potentials.