Approximating Multi-Dimensional Hamiltonian Flows by Billiards |
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Authors: | A Rapoport V Rom-Kedar D Turaev |
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Institution: | (1) Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, 76100, Israel;(2) Department of Mathematics, Ben-Gurion University, P.O.B. 653, Beèr Sheva, 84105, Israel |
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Abstract: | The behavior of a point particle traveling with a constant speed in a region , 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 , that, in the limit , become singular hard-wall potentials of multi-dimensional billiards. We define auxiliary billiard domains that asymptote,
as 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. |
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