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The nonholonomic bracket on contact mechanical systems

Manuel De León, Víctor M. Jiménez

Abstract

In this paper we study contact nonholonomic mechanical sys\-tems. We construct a general framework for non-holonomic constraints in contact geometry and, in this framework, we define different nonholonomic brackets using con\-venient \linebreak decompositions of the tangent bundle of the phase space. \linebreak Furthermore, we prove that all of them coincide. In particular, one the brackets is a natural extension of that defined by R.J. Eden, but now it is an almost Jacobi bracket since it is not satisfy the Leibniz rule.

The nonholonomic bracket on contact mechanical systems

Abstract

In this paper we study contact nonholonomic mechanical sys\-tems. We construct a general framework for non-holonomic constraints in contact geometry and, in this framework, we define different nonholonomic brackets using con\-venient \linebreak decompositions of the tangent bundle of the phase space. \linebreak Furthermore, we prove that all of them coincide. In particular, one the brackets is a natural extension of that defined by R.J. Eden, but now it is an almost Jacobi bracket since it is not satisfy the Leibniz rule.
Paper Structure (9 sections, 27 theorems, 213 equations)

This paper contains 9 sections, 27 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. Contact Hamiltonian systems
  3. General framework for contact Hamiltonian constrained dymanics
  4. Non-holonomic brackets
  5. The P-nonholonomic bracket
  6. Contact Lagrangian dynamics
  7. Constrained dymanics of contact Hamiltonian systems
  8. The contact Eden bracket
  9. Conclusions and further work

Key Result

Theorem 1

Given a manifold $M$ and a $\mathbb{R}$-bilinear map $\lbrace\cdot,\cdot\rbrace: \mathcal{C}^\infty(M) \times \mathcal{C}^{\infty}(M) \to \mathcal{C}^\infty(M)$, the following assertions are equivalent.

Theorems & Definitions (50)

  • Definition 1
  • Definition 2
  • Theorem 1: deLeon2019
  • Definition 3
  • Remark 1
  • Example 1
  • Theorem 2: Darboux theorem
  • Remark 2
  • Definition 4
  • Lemma 3: primero
  • ...and 40 more