Contact line bundles, foliations, and integrability

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.

19 pages, minor typos corrected

References in corpus (4)

• Torus Actions and Integrable Systems
• Completely Integrable Contact Hamiltonian Systems and Toric Contact Structures on $S^2\times S^3$
• Symmetries and Integrability
• Integrable systems in cosymplectic geometry