The Hamiltonian and Lagrangian approaches to the dynamics of nonholonomic systems
type: Item
instance of: scholarly article
title: The Hamiltonian and Lagrangian approaches to the dynamics of nonholonomic systems
zbMATH Open document ID: 0929.70009
DOI: 10.1016/S0034-4877(97)85617-0
author: Wang Sang Koon
author: J. E. Marsden
published in: Reports on Mathematical Physics
publication date: 2000-01-18T00:00:00Z
review text: The aim of this paper is to bring together two main approaches to nonholonomic mechanics: the Hamiltonian approach [see, for example, \textit{L. Bates} and \textit{J. Sniatycki}, Rep. Math. Phys. 32, No. 1, 99-115 (1993; Zbl 0798.58026)] and the Lagrangian approach [see \textit{A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden} and \textit{R. M. Murray}, Arch. Ration. Mech. Anal. 136, No. 1, 21-99 (1996; Zbl 0886.70014)]. Consider a configuration space \(Q\), a hyperregular Lagrangian \(L\), and a distribution \({\mathcal D}\) that describes the kinematic nonholonomic constraints. The authors construct a distribution \({\mathcal K}\) on the constraint submanifold \({\mathcal D}\) of \(TQ\). The solution of the constrained dynamics \(X_{\mathcal K}\) is given by the unique solution of equation \(i_{X_{\mathcal K}}\Omega_{K}=dE_{\mathcal K}\), where \(\Omega_{\mathcal K}\) and \(dE_{\mathcal K}\) are respectively the restrictions of the Poincaré-Cartan 2-form \(\Omega_L\) and the energy \(dE_L\) to \({\mathcal K}\) [see, also, \textit{M. de León, J. C. Marrero} and the reviewer, Int. J. Theor. Phys. 36, No. 4, 979-995 (1997; Zbl 0874.70012)]. Then, via the Legendre map, the authors obtain a precise relation between the constructions of the first two references. Now, it is added the hypothesis of symmetry, that is, it is assumed that a Lie group \(G\) acts on \(Q\) leaving the Lagrangian and the distribution invariant. It is proved that the reduced Lagrange-d'Alembert equations established in the first reference are equivalent to the reduced nonholonomic equations given in the second reference. As a nice application, the example of a simplified model of bicycle illustrates the results of the paper.
Mathematics Subject Classification ID: 70F25
Mathematics Subject Classification ID: 70H05
Mathematics Subject Classification ID: 70H03
Mathematics Subject Classification ID: 70G65
zbMATH DE Number: 1136528
zbMATH Keywords: configuration space
zbMATH Keywords: hyperregular Lagrangian
zbMATH Keywords: Poincaré-Cartan 2-form
zbMATH Keywords: Legendre map
zbMATH Keywords: symmetry
zbMATH Keywords: Lie group
zbMATH Keywords: reduced nonholonomic equations
reviewed by: David Martín de Diego
MaRDI profile type: Publication
cites work: Q4174811
cites work: Examples of gauge conservation laws in nonholonomic systems
cites work: Nonholonomic reduction
cites work: Nonholonomic mechanical systems with symmetry
cites work: Q4863254
cites work: Hamiltonian mechanics in the presence of constraints
cites work: A symmetric sphere rolling on a surface
cites work: Reduction of some classical non-holonomic systems with symmetry
cites work: Optimal Control for Holonomic and Nonholonomic Mechanical Systems with Symmetry and Lagrangian Reduction
cites work: Reduction of constrained mechanical systems and stability of relative equilibria
cites work: Q5594352
cites work: Reduction, symmetry, and phases in mechanics
cites work: Q3751065
cites work: Lagrangian reduction and the double spherical pendulum
cites work: On the Hamiltonian formulation of nonholonomic mechanical systems
cites work: Q4711784
cites work: Hamiltonian systems with constraints and their meaning in mechanics
full work available at URL: s0034-4877(97)85617-0
OpenAlex ID: W2149297753
Recommended article: Q4400547
Recommended article: Non-holonomic Lagrangian and Hamiltonian mechanics: an intrinsic approach
Recommended article: Nonholonomic reduction
Recommended article: Reduction of nonholonomic mechanical systems with symmetries
Recommended article: THE CARTAN FORM FOR CONSTRAINED LAGRANGIAN SYSTEMS AND THE NONHOLONOMIC NOETHER THEOREM
Recommended article: Q4425090
Recommended article: Mechanical systems with nonlinear constraints
Recommended article: Nonholonomic reduction for free and proper actions
Recommended article: The momentum equation and the second-order differential equation condition
Recommended article: GEOMETRIC HAMILTON–JACOBI THEORY FOR NONHOLONOMIC DYNAMICAL SYSTEMS
description: scientific article; zbMATH DE number 1136528