Contact complete integrability

TL;DR

This paper characterizes contact complete integrability through a structure involving Legendrian and co-Legendrian foliations, establishing its equivalence with known definitions and demonstrating solvability in quadratures with an example of billiards in pseudo-Euclidean space.

Contribution

It introduces a novel geometric framework for contact integrability using a flag of foliations and holonomy-invariant measures, connecting it to existing concepts and providing new solvability results.

Findings

Contact integrability characterized by a Legendrian-co-Legendrian foliation structure.

Contact completely integrable systems are solvable in quadratures.

Example of billiard system inside an ellipsoid in pseudo-Euclidean space demonstrating the theory.

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 $\R\ltimes \R^{n-1}$ structure on the leaves of the co-Legendrian foliation. Further, the above structure implies the existence of $n$ 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.