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 Regul. Chaotic Dyn., 2010, Volume 15, Issue 4-5, Pages 504–520 (Mi rcd512)

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

DOI: https://doi.org/10.1134/S1560354710040076

Bibliographic databases:

MSC: 37J35, 37J55, 70H06
Accepted:26.03.2010
Language:

Citation: B. Khesin, S. Tabachnikov, “Contact complete integrability”, Regul. Chaotic Dyn., 15:4-5 (2010), 504–520

Citation in format AMSBIB
\Bibitem{KheTab10} \by B. Khesin, S. Tabachnikov \paper Contact complete integrability \jour Regul. Chaotic Dyn. \yr 2010 \vol 15 \issue 4-5 \pages 504--520 \mathnet{http://mi.mathnet.ru/rcd512} \crossref{https://doi.org/10.1134/S1560354710040076} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2679761} \zmath{https://zbmath.org/?q=an:1203.37094} 

• http://mi.mathnet.ru/eng/rcd512
• http://mi.mathnet.ru/eng/rcd/v15/i4/p504

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Nguyen Tien Zung, “A Conceptual Approach to the Problem of Action-Angle Variables”, Arch. Ration. Mech. Anal., 229:2 (2018), 789–833
2. Jovanovic B., Jovanovic V., “Heisenberg Model in Pseudo-Euclidean Spaces II”, Regul. Chaotic Dyn., 23:4 (2018), 418–437
3. Sergyeyev A., “New Integrable (3+1)-Dimensional Systems and Contact Geometry”, Lett. Math. Phys., 108:2 (2018), 359–376