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Projective geodesic extensions by conformal modifications in nonholonomic mechanics

Malika Belrhazi, Tom Mestdag

Abstract

Projective geodesic extensions are reparametrizations of the trajectories of a nonholonomic mechanical system (with only a kinetic energy Lagrangian), in such a way that they can be interpreted as part of the geodesics of a Riemannian metric. We derive necessary and sufficient conditions for the existence of these extensions, in the case where the constrained Lagrangian remains preserved up to a conformal transformation. When the nonholonomic system has a symmetry group (a Chaplygin system), we clarify the relation between projective geodesic extensions and closely related concepts, such as $φ$-simplicity, invariant measures and Hamiltonization. Throughout the paper, new and relevant examples illustrate the key differences between all these concepts.

Projective geodesic extensions by conformal modifications in nonholonomic mechanics

Abstract

Projective geodesic extensions are reparametrizations of the trajectories of a nonholonomic mechanical system (with only a kinetic energy Lagrangian), in such a way that they can be interpreted as part of the geodesics of a Riemannian metric. We derive necessary and sufficient conditions for the existence of these extensions, in the case where the constrained Lagrangian remains preserved up to a conformal transformation. When the nonholonomic system has a symmetry group (a Chaplygin system), we clarify the relation between projective geodesic extensions and closely related concepts, such as -simplicity, invariant measures and Hamiltonization. Throughout the paper, new and relevant examples illustrate the key differences between all these concepts.

Paper Structure

This paper contains 23 sections, 25 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Purely kinetic nonholonomic systems
  4. Chaplygin systems
  5. Projective geodesic extensions
  6. The two principal conditions
  7. The generalized nonholonomic particle
  8. The two-wheeled carriage
  9. Projective geodesic extensions for Chaplygin systems
  10. The formulation on the configuration manifold
  11. The generalized nonholonomic particle
  12. $\varphi$-simplicity
  13. Relation to projective geodesic extensions on the configuration manifold
  14. The generalized nonholonomic particle
  15. The two-wheeled carriage
  16. ...and 8 more sections

Key Result

Lemma 1

Let $g$ be a Riemannian metric and let $\hat{g}$ be a pseudo-Riemannian metric which is a $\mathcal{D}$-conformal modification of $g$, $\hat{g}=e^{2F}\overline{g}$. Assume that ${\overline g}|_{\mathcal{D}^{\overline g} \times \mathcal{D}^{\overline g}}$ is non-degenerate, where $\mathcal{D}^{\overl where $\theta_i=\overline{g}_{ai}v^a$ and $\lambda_i$ are the Lagrange multipliers (lambda).

Theorems & Definitions (49)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 2
  • ...and 39 more