Completely Integrable Contact Hamiltonian Systems and Toric Contact Structures on $S^2\times S^3$

TL;DR

This paper explores completely integrable contact Hamiltonian systems, focusing on toric contact structures on S^2×S^3, and provides a classification of certain toric contact structures, resolving their contact equivalence problem.

Contribution

It offers a complete solution to the contact equivalence problem for the Y^{p,q} toric contact structures on S^2×S^3, establishing when they are contact inequivalent.

Findings

Y^{p,q} and Y^{p',q'} are contact inequivalent iff p ≠ p'

Provides a classification of toric contact structures on S^2×S^3

Advances understanding of integrable contact Hamiltonian systems

Abstract

I begin by giving a general discussion of completely integrable Hamiltonian systems in the setting of contact geometry. We then pass to the particular case of toric contact structures on the manifold $S^2\times S^3$. In particular we give a complete solution to the contact equivalence problem for a class of toric contact structures, $Y^{p,q}$, discovered by physicists by showing that $Y^{p,q}$ and $Y^{p',q'}$ are inequivalent as contact structures if and only if $p\neq p'$.