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This article is cited in 14 scientific papers (total in 14 papers)
On the 60th birthday of professor V.V. Kozlov
Contact complete integrability
B. Khesina, S. Tabachnikovb a Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada
b Department of Mathematics, Pennsylvania State University,
University Park, PA 16802, USA
Abstract:
Complete integrability in a symplectic setting means the existence of a Lagrangian foliation leaf-wise preserved by the dynamics. In the paper we describe complete integrability in a contact set-up as a more subtle structure: a flag of two foliations, Legendrian and co-Legendrian, and a holonomy-invariant transverse measure of the former in the latter. This turns out to be equivalent to the existence of a canonical $\mathbb{R}\times \mathbb{R}^{n-1}$ structure on the leaves of the co-Legendrian foliation. Further, the above structure implies the existence of n commuting contact fields preserving a special contact 1-form, thus providing the geometric framework and establishing equivalence with previously known definitions of contact integrability. We also show that contact completely integrable systems are solvable in quadratures.
We present an example of contact complete integrability: the billiard system inside an ellipsoid in pseudo-Euclidean space, restricted to the space of oriented null geodesics. We describe a surprising acceleration mechanism for closed light-like billiard trajectories.
Keywords:
complete integrability, contact structure, Legendrian foliation, pseudo-Euclidean geometry, billiard map.
Received: 02.10.2009 Accepted: 26.03.2010
Citation:
B. Khesin, S. Tabachnikov, “Contact complete integrability”, Regul. Chaotic Dyn., 15:4-5 (2010), 504–520
Linking options:
https://www.mathnet.ru/eng/rcd512 https://www.mathnet.ru/eng/rcd/v15/i4/p504
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