Contact line bundles, foliations, and integrability

TL;DR

This paper explores the concept of non-commutative integrability in contact systems on manifolds, utilizing Jacobi structures to unify cooriented and non-cooriented cases, including dissipative Hamiltonian systems.

Contribution

It introduces a formulation of non-commutative integrability for contact systems using Jacobi structures, applicable to both cooriented and non-cooriented contact manifolds.

Findings

Unified treatment of contact systems with and without global contact forms.

Extension of integrability concepts to dissipative Hamiltonian systems.

Reduction of Jacobi structures to standard forms in special cases.

Abstract

We formulate the non-commutative integrability of contact systems on a contact manifold $(M,\mathcal H)$ using the Jacobi structure on the space of sections $\Gamma(L)$ of a contact line bundle $L$. In the cooriented case, if the line bundle is trivial and $\mathcal H$ is the kernel of a globally defined contact form $\alpha$, the Jacobi structure on the space of sections reduces to the standard Jacobi structure on $(M,\alpha)$. We therefore treat contact systems on cooriented and non-cooriented contact manifolds simultaneously. In particular, this allows us to work with dissipative Hamiltonian systems where the Hamiltonian does not have to be preserved by the Reeb vector field.