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A new perspective on nonholonomic brackets and Hamilton-Jacobi theory

Manuel de León, Manuel Lainz, Asier López-Gordón, Juan Carlos Marrero

Abstract

The nonholonomic dynamics can be described by the so-called nonholonomic bracket on the constrained submanifold, which is a non-integrable modification of the Poisson bracket of the ambient space, in this case, of the canonical bracket on the cotangent bundle of the configuration manifold. On the other hand, another bracket, also called nonholonomic bracket, was defined using the description of the problem in terms of skew-symmetric algebroids. Recently, reviewing two older papers by R. J. Eden, we have defined a new bracket which we call Eden bracket. In the present paper, we prove that these three brackets coincide. Moreover, the description of the nonholonomic bracket à la Eden has allowed us to make important advances in the study of Hamilton-Jacobi theory and the quantization of nonholonomic systems.

A new perspective on nonholonomic brackets and Hamilton-Jacobi theory

Abstract

The nonholonomic dynamics can be described by the so-called nonholonomic bracket on the constrained submanifold, which is a non-integrable modification of the Poisson bracket of the ambient space, in this case, of the canonical bracket on the cotangent bundle of the configuration manifold. On the other hand, another bracket, also called nonholonomic bracket, was defined using the description of the problem in terms of skew-symmetric algebroids. Recently, reviewing two older papers by R. J. Eden, we have defined a new bracket which we call Eden bracket. In the present paper, we prove that these three brackets coincide. Moreover, the description of the nonholonomic bracket à la Eden has allowed us to make important advances in the study of Hamilton-Jacobi theory and the quantization of nonholonomic systems.
Paper Structure (21 sections, 13 theorems, 201 equations)

This paper contains 21 sections, 13 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Lagrangian and Hamiltonian mechanics: a brief survey
  3. Lagrangian mechanics
  4. Legendre transformation
  5. Hamiltonian description
  6. Nonholonomic mechanical systems
  7. The Lagrangian description
  8. The Hamiltonian description
  9. The skew-symmetric algebroid approach
  10. The nonholonomic bracket
  11. Eden bracket
  12. Comparison of brackets
  13. Application to the Hamilton--Jacobi theory
  14. Hamilton--Jacobi theory for standard Hamiltonian systems
  15. Hamilton--Jacobi theory for nonholonomic mechanical systems
  16. ...and 6 more sections

Key Result

Theorem 1

The vector bundle isomorphism over the identity of $Q$, given by the composition is an almost Poisson isomorphism between the almost Poisson manifolds $(M, \{\, , \, \}_{E})$ and $(D^*, \{\, , \, \}_{D^*})$ .

Theorems & Definitions (32)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Theorem 1
  • proof
  • Remark 7
  • Proposition 2
  • ...and 22 more