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Approximating multi-dimensional Hamiltonian flows by billiards

A. Rapoport·V. Rom-Kedar·D. Turaev
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Abstract

Consider a family of smooth potentials $V_ε$, which, in the limit $ε\to0$, become a singular hard-wall potential of a multi-dimensional billiard. We define auxiliary billiard domains that asymptote, as $ε\to0$ to the original billiard, and provide 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 which limit to the multi-dimensional close to ellipsoidal billiards, we predict when the separatrix splitting persists for various types of potentials.

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A. Rapoport, V. Rom-Kedar, D. Turaev. 2005-11-30. Approximating multi-dimensional Hamiltonian flows by billiards. https://doi.org/10.1007/s00220-007-0228-0

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