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Lagrangian reduction of nonholonomic discrete mechanical systems

Javier Fernandez, Cora Tori, Marcela Zuccalli

Abstract

In this paper we propose a process of lagrangian reduction and reconstruction for nonholonomic discrete mechanical systems where the action of a continuous symmetry group makes the configuration space a principal bundle. The result of the reduction process is a discrete dynamical system that we call the discrete reduced system. We illustrate the techniques by analyzing two types of discrete symmetric systems where it is possible to go further and obtain (forced) discrete mechanical systems that determine the dynamics of the discrete reduced system.

Lagrangian reduction of nonholonomic discrete mechanical systems

Abstract

In this paper we propose a process of lagrangian reduction and reconstruction for nonholonomic discrete mechanical systems where the action of a continuous symmetry group makes the configuration space a principal bundle. The result of the reduction process is a discrete dynamical system that we call the discrete reduced system. We illustrate the techniques by analyzing two types of discrete symmetric systems where it is possible to go further and obtain (forced) discrete mechanical systems that determine the dynamics of the discrete reduced system.

Paper Structure

This paper contains 32 sections, 35 theorems, 165 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Reduction of classical mechanical systems
  3. Generalized nonholonomic mechanical systems
  4. Symmetric generalized nonholonomic mechanical systems
  5. Reduction of symmetry
  6. Discrete mechanical systems and symmetries
  7. Discrete mechanical systems
  8. Symmetric discrete mechanical systems
  9. Some discrete tools
  10. Affine discrete connections
  11. Isomorphisms associated to an affine discrete connection
  12. Variations and reduced variations
  13. Reduced lagrangians
  14. Mixed curvature and reduced forces
  15. Reduced dynamics
  16. ...and 17 more sections

Key Result

Theorem 2.3

A smooth curve $q$ in $Q$ is a trajectory of $(Q,L,\mathcal{D},C_K)$ if and only if $(q(t),\dot{q}(t))\in C_K$ for all $t\in [t_0,t_1]$ and where $\mathbb{F} L$ and $\mathbb{B} L$ denote the fiber and base derivatives of $L$, as defined in ar:cendra_ferraro_grillo-lagrangian_reduction_of_generalized_NHS and $(\mathcal{D}_{q(t)})^\circ\subset T^*_{q(t)}Q$ is the annihilator of $\mathcal{D}_{q(t)}\

Figures (1)

  • Figure 1: $G$ orbit and $Hor_{{\mathcal{A}_d}}^2(q_0)$ in $Q$

Theorems & Definitions (121)

  • Definition 2.1
  • Definition 2.2: Lagrange--D'Alembert Principle
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6: Generalized Lagrange--D'Alembert--Poincaré Principle
  • Theorem 2.7
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3: Discrete Lagrange--D'Alembert Principle
  • ...and 111 more