Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Coisotropic embeddings of precosymplectic manifolds

Manuel de León, Pablo Soto Martín

Abstract

In this paper we provide a complete characterisation of coisotropic embeddings of precosymplectic manifolds into cosymplectic manifolds. This result extends a theorem of Gotay about coisotropic embeddings of presymplectic manifolds. We also extend to the cosymplectic case some results of A. Weinstein which generalise the Darboux theorem. While symplectic geometry is the natural framework for developing Hamiltonian mechanics, cosymplectic geometry is the corresponding framework for time-dependent Hamiltonian mechanics. The motivation behind proving this theorem is to generalise known results for symplectic geometry to cosymplectic geometry, so that they can be used to study time-dependent systems, for instance for the regularization problem of singular Lagrangian systems.

Coisotropic embeddings of precosymplectic manifolds

Abstract

In this paper we provide a complete characterisation of coisotropic embeddings of precosymplectic manifolds into cosymplectic manifolds. This result extends a theorem of Gotay about coisotropic embeddings of presymplectic manifolds. We also extend to the cosymplectic case some results of A. Weinstein which generalise the Darboux theorem. While symplectic geometry is the natural framework for developing Hamiltonian mechanics, cosymplectic geometry is the corresponding framework for time-dependent Hamiltonian mechanics. The motivation behind proving this theorem is to generalise known results for symplectic geometry to cosymplectic geometry, so that they can be used to study time-dependent systems, for instance for the regularization problem of singular Lagrangian systems.

Paper Structure

This paper contains 12 sections, 12 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Cosymplectic geometry
  3. Coisotropic embeddings
  4. Coisotropic embeddings of precosymplecitc manifolds
  5. Direct theorem
  6. Existence theorem
  7. Local Uniqueness theorem
  8. Conclusions and future work
  9. Coisotropic embeddings of presymplectic manifolds
  10. Direct theorem
  11. Existence theorem
  12. Local uniqueness theorem

Key Result

Theorem 3.1

Every coisotropic submanifold of a cosymplectic manifold has an induced precosymplectic structure. Additionally, every precosymplectic manifold can be coisotropically embedded into a cosymplectic manifold. This embedding is unique up to a local cosymplectomorphism.

Theorems & Definitions (24)

  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • proof
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • ...and 14 more