Contactifications: a Lagrangian description of compact Hamiltonian systems

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 $N=M\mathrm{/}\mathcal{F}$. We call $(M,\eta )$ a $\mathit{\text{c}}ontactification$ 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’.