Contactifications: a Lagrangian description of compact Hamiltonian systems

If η is a contact form on a manifold M such that the orbits of the Reeb vector field form a simple foliation F on M, then the presymplectic 2-form dη on M induces a symplectic structure ω on the quotient manifold N=M/F. We call (M,η) a contactification of the symplectic manifold (N,ω). 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 CPⁿ⁻¹, being the unit sphere S²ⁿ⁻¹ in Cⁿ, and equipped with the restriction of the Liouville 1-form on Cⁿ. 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,ω), on which symplectic forms never admit a `vector potential'.