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.

33 pages, some corrections in Section 5

References in corpus (25)

• Contact Hamiltonian Mechanics
• Higher vector bundles and multi-graded symplectic manifolds
• Contact Hamiltonian Systems
• Geometry of Thermodynamic Processes
• New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetries
• A Hamilton-Jacobi Formalism for Thermodynamics
• Graded contact manifolds and contact Courant algebroids
• Contact manifolds and dissipation, classical and quantum
• The graded Jacobi algebras and (co)homology
• Singular Lagrangians and precontact Hamiltonian Systems
• A novel approach to contact Hamiltonians and contact Hamilton-Jacobi theory
• Quasi-derivations and QD-algebroids
• Remarks on Contact and Jacobi Geometry
• Linear duals of graded bundles and higher analogues of (Lie) algebroids
• Brackets
• Contact fiber bundles
• Unified Lagrangian-Hamiltonian formalism for contact systems
• Chekanov's dichotomy in contact topology
• Time-dependent Mechanics and Lagrangian submanifolds of Dirac manifolds
• Poisson-Jacobi reduction of homogeneous tensors
• Rigidity of integral coisotropic submanifolds of contact manifolds
• Jacobi bundles and the BFV-complex
• Symmetries and Covariant Poisson brackets on pre-symplectic manifolds
• Covariant Variational Evolution and Jacobi Brackets: Fields
• Covariant Variational Evolution and Jacobi Brackets: Particles

Cited by in corpus (4)

• Contact Lie systems
• Contactifications: a Lagrangian description of compact Hamiltonian systems
• Liouville-Arnold theorem for homogeneous symplectic and contact Hamiltonian systems
• Jacobi algebroids and Jacobi sigma models