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Boris Khesin

40 St. George Street, Toronto, ON M5S 2E4, Canada
Department of Mathematics, University of Toronto


Khesin B. A., Tabachnikov S.
Contact complete integrability
2010, vol. 15, no. 4-5, pp.  504-520
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
Citation: Khesin B. A., Tabachnikov S.,  Contact complete integrability, Regular and Chaotic Dynamics, 2010, vol. 15, no. 4-5, pp. 504-520

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