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Reductions: precontact versus presymplectic

Katarzyna Grabowska, Janusz Grabowski

TL;DR

This work presents an intrinsic, geometrical framework for reducing systems with symmetry in the context of general contact and precontact structures. By viewing contact manifolds as homogeneous symplectic manifolds on symplectic ${\mathbb{R}^\times}$-bundles, it lifts group actions to Hamiltonian ones on the cover and defines 1-homogeneous Hamiltonians, enabling moment maps defined on the cover rather than on the base manifold. The authors develop a precontact-to-contact reduction, a precontact Marsden-Weinstein-Meyer theorem, and a constant-rank reduction theory, all formulated in terms of presymplectic covers and their foliations, with clear reduction diagrams that commute with symmetry actions. This framework unifies and extends traditional contact reductions, accommodating non-coorientable structures and presymplectic settings, and has potential applications to constrained and covariant physical theories. The approach provides a robust, intrinsic toolkit for reductions that can be directly applied to complex systems in physics and geometry.

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.

Reductions: precontact versus presymplectic

TL;DR

This work presents an intrinsic, geometrical framework for reducing systems with symmetry in the context of general contact and precontact structures. By viewing contact manifolds as homogeneous symplectic manifolds on symplectic -bundles, it lifts group actions to Hamiltonian ones on the cover and defines 1-homogeneous Hamiltonians, enabling moment maps defined on the cover rather than on the base manifold. The authors develop a precontact-to-contact reduction, a precontact Marsden-Weinstein-Meyer theorem, and a constant-rank reduction theory, all formulated in terms of presymplectic covers and their foliations, with clear reduction diagrams that commute with symmetry actions. This framework unifies and extends traditional contact reductions, accommodating non-coorientable structures and presymplectic settings, and has potential applications to constrained and covariant physical theories. The approach provides a robust, intrinsic toolkit for reductions that can be directly applied to complex systems in physics and geometry.

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.
Paper Structure (17 sections, 28 theorems, 110 equations)

This paper contains 17 sections, 28 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Principles of precontact geometry
  3. Presymplectic ${\mathbb{R}^\times}$-bundles
  4. Remarks on Poisson ${\mathbb{R}^\times}$-bundles
  5. Precontact-to-contact reduction
  6. Symplectic reduction of presymplectic ${\mathbb{R}^\times}$-bundles
  7. Precontact and presymplectic reductions
  8. Hamiltonian dynamics on precontact manifolds
  9. Precontact Hamiltonians
  10. Contact Hamiltonian mechanics
  11. Contact reductions
  12. Isotropic, coisotropic, and Legendrian submanifolds
  13. Constant rank reduction
  14. Contact Marsden-Weinstein-Meyer reduction
  15. Precontact moment maps
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $(M,C)$ be a contact manifold with a symplectic cover $焜:P\to M$, let $焚:G\times M\to M$ be an action on $M$ of a Lie group $G$ by contactomorphisms, and let $J:P\to\mathfrak{g}^*$ be the corresponding contact moment map. Let $焖\in\mathfrak{g}^*$ be a weakly regular value of $J$, so the connecte of the Lie algebra $\mathfrak{g}$ of $G$, acts on the submanifold $M_焖=焜(J^{-1}(焖))$ of $M$. In par

Theorems & Definitions (62)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Theorem 2.4: Precontact Darboux Theorem
  • proof
  • Remark 2.5
  • Definition 2.6
  • Proposition 2.7
  • ...and 52 more