Contact complete integrability

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 $\mathbb{R}⋉{\mathbb{R}}^{n-1}$ 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.

References in corpus (1)

• On Elliptical Billiards in the Lobachevsky Space and associated Geodesic Hierarchies

Cited by in corpus (11)

• Completely Integrable Contact Hamiltonian Systems and Toric Contact Structures on $S^2\times S^3$
• Contact flows and integrable systems
• Quasi-morphisms on contactomorphism groups and contact rigidity
• Hamilton-Jacobi theory and integrability for autonomous and non-autonomous contact systems
• Heisenberg model in pseudo-Euclidean spaces
• Integrable systems in cosymplectic geometry
• Heisenberg model in pseudo-Euclidean spaces II
• Liouville-Arnold theorem for homogeneous symplectic and contact Hamiltonian systems
• Bott-integrable Reeb flows on 3-manifolds
• On a Trivial Family of Noncommutative Integrable Systems
• Contact line bundles, foliations, and integrability