Reductions: precontact versus presymplectic

Abstract

We show that contact reductions can be described in terms of symplectic reductions in the traditional Marsden–Weinstein–Meyer as well as the constant rank picture. The point is that we view contact structures as particular (homogeneous) symplectic structures. A group action by contactomorphisms is lifted to a Hamiltonian action on the corresponding symplectic manifold, called the symplectic cover of the contact manifold. In contrast to the majority of the literature in the subject, our approach includes general contact structures (not only co-oriented) and changes the traditional view point: contact Hamiltonians and contact moment maps for contactomorphism groups are no longer defined on the contact manifold itself, but on its symplectic cover. Actually, the developed framework for reductions is slightly more general than purely contact, and includes a precontact and presymplectic setting which is based on the observation that there is a one-to-one correspondence between isomorphism classes of precontact manifolds and certain homogeneous presymplectic manifolds.

Contact Lie systems: theory and applications

“…It is convenient to state the following contact Marsden-Weinstein reduction theorem for contact Hamiltonian systems. Although it follows immediately from previous comments and theories [3,47,49,81], it seems to be absent in the literature.…”

“…Contact forms, in particular, and contact geometry, in general, have proved to be very useful in many different problems in areas such as thermodynamics [14,75], circuit theory [46], non-holonomic systems [28], quantum mechanics [24], gravitation and general relativity [44,65], control theory [66], among others [32,57,77]. Moreover, contact geometry has drawn, by itself, much attention in recent times [47][48][49]. Recently, the notion of cocontact manifold has also been developed to introduce explicit dependence on time [25,43,70].…”