The momentum equation and the second-order differential equation condition
type: Item
instance of: scholarly article
title: The momentum equation and the second-order differential equation condition
zbMATH Open document ID: 1014.70015
DOI: 10.1016/S0034-4877(02)80034-9
published in: Reports on Mathematical Physics
publication date: 2003-01-22T00:00:00Z
review text: The equations of motion of mechanical systems with nonholonomic constraints are investigated. It is shown that the momentum equation is valid for a larger class of functions than the nonholonomic momenta of the article [\textit{A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden}, and \textit{R. M. Murray}, Arch. Ration. Mech. Anal. 136, 21--99 (1996; Zbl 0886.70014)]. It is proved that the equations of motion are uniquely characterized as the second-order differential equations compatible with the momentum equation. For a free and proper action of a symmetry group \(G\), the reduced equations on the orbit space are obtained by restricting the second-order differential equation condition and the momentum equation to \(G\)- invariant functions and vector fields respectively. This practically important approach is illustrated by a simple derivation of the reduced equations of motion in the case when the configuration space coincides with the group manifold of the symmetry group \(G\), and the action of \(G\) on itself is given by the left multiplication. If the action of \(G\) is proper but not free, the restriction of the second-order differential equation condition and the momentum equation to \(G\)-invariant functions does not give equations of motion on the whole constraint distribution \(D\), but on an open subset of \(D\). This situation is also illustrated by an example showing the limitations of the method.
Mathematics Subject Classification ID: 70H33
Mathematics Subject Classification ID: 70G65
Mathematics Subject Classification ID: 37J15
Mathematics Subject Classification ID: 37J60
Mathematics Subject Classification ID: 53D20
Mathematics Subject Classification ID: 70G45
Mathematics Subject Classification ID: 70F25
zbMATH DE Number: 1857371
zbMATH Keywords: Hamiltonian mechanics
zbMATH Keywords: nonholonomic constraints
zbMATH Keywords: reduction of symmetries
Wikidata QID: Q115339956
author: J. Śniatycki
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
cites work: Q3976412
cites work: Symmetry and bifurcations of momentum mappings
cites work: Proper group actions and symplectic stratified spaces
cites work: Nonholonomic reduction
cites work: Nonholonomic mechanical systems with symmetry
cites work: Q2762109
cites work: Q2785808
cites work: Q3647601
cites work: Q4266451
cites work: Differential structure of orbit spaces.
cites work: Lie groups
cites work: Q4039743
cites work: Q5731354
cites work: Various approaches to conservative and nonconservative nonholonomic systems
cites work: Stratified symplectic spaces and reduction
cites work: Almost Poisson spaces and nonholonomic singular reduction
cites work: Nonholonomic Noether theorem and reduction of symmetries
cites work: Accessible Sets, Orbits, and Foliations with Singularities
cites work: Orbits of Families of Vector Fields and Integrability of Distributions
Recommended article: Reduction of a Hamilton-Jacobi equation for nonholonomic systems
Recommended article: Q3647601
Recommended article: Reduction and reconstruction of the dynamics of nonholonomic systems
Recommended article: Reduction of constrained mechanical systems and stability of relative equilibria
Recommended article: Reduction of nonholonomic mechanical systems with symmetries
Recommended article: Q5442450
Recommended article: Nonholonomic reduction for free and proper actions
Recommended article: Hamiltonian formulation for mechanical systems with nonholonomic constraints
Recommended article: Reduction of invariant constrained systems using anholonomic frames
Recommended article: Q3826145
description: scientific article; zbMATH DE number 1857371