Discrete Lagrange-d'Alembert-Poincaré equations for Euler's disk
type: Item
instance of: scholarly article
title: Discrete Lagrange-d'Alembert-Poincaré equations for Euler's disk
zbMATH Open document ID: 1279.37042
author: Cédric M. Campos
author: Viviana Alejandra Díaz
author: David Martín de Diego
author: Hernan Cendra
published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM
publication date: 2013-01-30T00:00:00Z
review text: A nonholonomic connection \(\mathcal{A}\) on a principal \(G\)-bundle \(Q\) which is equipped with a \(G\)-invariant metric and a distribution \(\mathcal{D}\) such that \(TQ=\mathcal{D}+\mathcal{V}\) for the vertical distribution \(\mathcal{V}\) is characterized by the property that its horizontal tangent distribution is the orthogonal complement of the distribution \(\mathcal{S}:=\mathcal{D} \cap\mathcal{V}\) in \(\mathcal{D}\) at every point of \(Q\). Such a nonholonomic connection \(\mathcal{A}\) gives a vector bundle isomorphism \(\alpha _{\mathcal{A}}:\left( TQ\right) /G\rightarrow T\left( Q/G\right) \oplus\widetilde{\mathfrak{g}}\) for the adjoint bundle \(\widetilde{\mathfrak{g}}\) of the principal \(G\)-bundle, providing a reduced framework in which the variational Lagrange-d'Alembert principle can be formulated in a reduced form of Lagrange-d'Alembert-Poincaré equations. After some discretization process, this is applied to derive a system of reduced discrete equations for the study of Euler's disk, where a thick disk (or a thin cylinder) with a rotationally symmetric mass distribution with respect to its central axis is rolling without sliding on a rough surface with only one point of contact. The variational integrator obtained in the discretization of Lagrange-d'Alembert-Poincaré equations for this nonholonomic system is numerically simulated. The result shows that the contact point's path is closed and the energy level remains almost constant, in significant contrast with some decaying phenomena found in the simulation by the Runge-Kutta method.
Mathematics Subject Classification ID: 37J60
Mathematics Subject Classification ID: 53D17
Mathematics Subject Classification ID: 53D20
Mathematics Subject Classification ID: 70F25
Mathematics Subject Classification ID: 70H45
zbMATH DE Number: 6132863
zbMATH Keywords: Euler's disk
zbMATH Keywords: discrete Lagrange-d'Alembert-Poincaré equations
zbMATH Keywords: Lagrange-d'Alembert principle
zbMATH Keywords: variational integrator
zbMATH Keywords: nonholonomic connection
zbMATH Keywords: Lagrangian system
reviewed by: Albert Sheu
MaRDI profile type: Publication
full work available at URL: s13398-011-0053-3
OpenAlex ID: W1999379885
cites work: Nonholonomic mechanics and control. With the collaboration of J. Baillieul, P. Crouch, and J. Marsden. With scientific input from P. S. Krishnaprasad, R. M. Murray, and D. Zenkov.
cites work: Nonholonomic mechanical systems with symmetry
cites work: Dynamics of rolling disk
cites work: Q2762109
cites work: The Lagrange-d'Alembert-Poincaré equations and integrability for the rolling disk
cites work: The Lagrange-D'Alembert-Poincaré equations and integrability for the Euler's disk
cites work: Geometric control and numerical aspects of nonholonomic systems
cites work: Non-holonomic integrators
cites work: Q3647601
cites work: Q4863254
cites work: Geometric integrators and nonholonomic mechanics
cites work: Discrete nonholonomic LL systems on Lie groups
cites work: Dynamics of the discrete Chaplygin sleigh
cites work: Q4531870
cites work: Discrete nonholonomic Lagrangian systems on Lie groupoids
cites work: The ringing of Euler's disk
cites work: Discrete mechanics and variational integrators
cites work: Integrators for nonholonomic mechanical systems
cites work: Nonlinear oscillations in the rolling motion of Euler's disk
DOI: 10.1007/S13398-011-0053-3
description: scientific article; zbMATH DE number 6132863