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'.
Categories: math.SG math-ph math.MP