Contactifications: a Lagrangian description of compact Hamiltonian systems

TL;DR

This paper introduces the concept of contactification as a Lagrangian framework for Hamiltonian systems, generalizing known constructions and providing new geometric tools for symplectic reduction and analysis.

Contribution

It presents explicit constructions of contactifications for coadjoint orbits, generalizes classical examples, and offers a method to analyze Hamiltonian systems on compact symplectic manifolds.

Findings

Explicit geometric construction of contactifications for coadjoint orbits

Generalization of contactification of complex projective space

Application of contactifications to Lagrangian description of Hamiltonian systems

Abstract

If $\eta$ is a contact form on a manifold $M$ such that the orbits of the Reeb vector field form a simple foliation $\mathcal{F}$ on $M$, then the presymplectic 2-form $d\eta$ on $M$ induces a symplectic structure $\omega$ on the quotient manifold $N=M/\mathcal{F}$. We call $(M,\eta)$ a $\textit contactification$ of the symplectic manifold $(N,\omega)$. First, we present an explicit geometric construction of contactifications of some coadjoint orbits of connected Lie groups. Our construction is a far going generalization of the well-known contactification of the complex projective space $\mathbb{C}P^{n-1}$, being the unit sphere $S^{2n-1}$ in $\mathbb{C}^{n}$, and equipped with the restriction of the Liouville 1-form on $\mathbb{C}^n$. Second, we describe a constructive procedure for obtaining contactification in the process of the Marsden-Weinstein-Meyer symplectic reduction and indicate geometric obstructions for the existence of compact contactifications. Third, we show that contactifications provide a nice geometrical tool for a Lagrangian description of Hamiltonian systems on compact symplectic manifolds $(N,\omega)$, on which symplectic forms never admit a `vector potential'.