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Regularization of singular time-dependent Lagrangian systems

Manuel De León, Rubén Izquierdo-López, Luca Schiavone, Pablo Soto

Abstract

One approach to studying the dynamics of a singular Lagrangian system is to attempt to regularize it, that is, to find an equivalent and regular system. In the case of time-independent singular Lagrangians, an approach due to \textit{A. Ibort} and \textit{J. Marín-Solano} is to use the coisotropic embedding theorem proved by \textit{M.J. Gotay} which states that any pre-symplectic manifold can be coisotropically embedded in a symplectic manifold. In this paper, we revisit these results and provide an alternative approach, also based on the coisotropic embedding theorem, that employs the Tulczyjew isomorphism and almost product structures, and allows for a slight generalization of the construction. In this revision, we also prove uniqueness of the Lagrangian regularization to first order. Furthermore, we extend our methodology to the case of time-dependent singular Lagrangians.

Regularization of singular time-dependent Lagrangian systems

Abstract

One approach to studying the dynamics of a singular Lagrangian system is to attempt to regularize it, that is, to find an equivalent and regular system. In the case of time-independent singular Lagrangians, an approach due to \textit{A. Ibort} and \textit{J. Marín-Solano} is to use the coisotropic embedding theorem proved by \textit{M.J. Gotay} which states that any pre-symplectic manifold can be coisotropically embedded in a symplectic manifold. In this paper, we revisit these results and provide an alternative approach, also based on the coisotropic embedding theorem, that employs the Tulczyjew isomorphism and almost product structures, and allows for a slight generalization of the construction. In this revision, we also prove uniqueness of the Lagrangian regularization to first order. Furthermore, we extend our methodology to the case of time-dependent singular Lagrangians.

Paper Structure

This paper contains 29 sections, 21 theorems, 239 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Distributions and foliations
  4. A Tulczyjew isomorphism for foliations
  5. Tangent structures
  6. Jet structures
  7. Regularization of autonomous systems
  8. Symplectic Hamiltonian systems
  9. Coisotropic regularization of pre-symplectic Hamiltonian systems
  10. Inconsistent Hamiltonian Systems.
  11. Consistent Hamiltonian Systems.
  12. Coisotropic regularization of degenerate Lagrangian systems
  13. Inconsistent Lagrangian Systems.
  14. Consistent Lagrangian Systems.
  15. Existence and uniqueness of Lagrangian regularization
  16. ...and 14 more sections

Key Result

Theorem 1

A regular distribution $D$ on a smooth manifold $Q$ is integrable if and only if it is involutive.

Theorems & Definitions (96)

  • Definition 1: Regular Distribution
  • Definition 2: Involutivity and Integrability
  • Remark 1
  • Theorem 1: Frobenius Theorem
  • Definition 3: Regular Foliation
  • Remark 2: Relation to Distributions
  • Definition 4: Tangent Bundle of a Foliation
  • Definition 5: Cotangent Bundle of a Foliation
  • Remark 3
  • Remark 4
  • ...and 86 more