Contactifications: a Lagrangian description of compact Hamiltonian systems*

Abstract

Abstract If η is a contact form on a manifold M such that the orbits of the Reeb vector field R form a simple foliation F on M, then the presymplectic 2-form d η 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 C P n − 1 , being the unit sphere S 2 n − 1 in C n , and equipped with the restriction of the Liouville 1-form on 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 , ω ) , on which symplectic forms never admit a ‘vector potential’.