Differential geometry  of rigid bodies collisions and non-standard billiards

Christopher  Cox; Renato  Feres

doi:10.3934/dcds.2016065

Article Contents

2016, Volume 36, Issue 11: 6065-6099. Doi: 10.3934/dcds.2016065

This issue Previous Article Second-order variational problems on Lie groupoids and optimal control applications Next Article The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations

Differential geometry of rigid bodies collisions and non-standard billiards

Christopher Cox^1, and
Renato Feres^1,

1.
Washington University, Department of Mathematics, One Brookings Dr., Campus Box 1146, St. Louis, MO 63130

Received: September 30, 2015

Revised: May 31, 2016

Published: May 2017

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Abstract

The configuration manifold $M$ of a mechanical system consisting of two unconstrained rigid bodies in $\mathbb{R}^n$ is a manifold with boundary (typically with singularities.) A full description of the system requires boundary conditions specifying how orbits should be continued after collisions, that is, the assignment of a collision map at each tangent space on the boundary of $M$ giving the post-collision state of the system for each pre-collision state. We give a complete description of the space of linear collision maps satisfying energy and (linear and angular) momentum conservation, time reversibility, and the natural requirement that impulse forces only act at the point of contact of the colliding bodies. These assumptions are stated geometrically in terms of a family of vector subbundles of the tangent bundle to $\partial M$: the diagonal, non-slipping, and impulse subbundles. Collision maps are shown to be the isometric involutions that restrict to the identity on the non-slipping subspace. We then make a few observations of a dynamical nature about non-standard billiard systems, among which is a sufficient condition for the billiard map on the space of boundary states to preserve the canonical measure on constant energy hypersurfaces.

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Mathematics Subject Classification: Primary: 37F35; Secondary: 37D50.

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