Contact complete integrability

B. Khesin, S. Tabachnikov

Research output: Contribution to journalArticlepeer-review

14 Scopus citations

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 ℝ ⋉ ℝn-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.

Original languageEnglish (US)
Pages (from-to)504-520
Number of pages17
JournalRegular and Chaotic Dynamics
Volume15
Issue number4
DOIs
StatePublished - 2010

All Science Journal Classification (ASJC) codes

  • Mathematics (miscellaneous)

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