Examples of singular nonholonomic reduction
type: Item
instance of: scholarly article
title: Examples of singular nonholonomic reduction
zbMATH Open document ID: 0931.37022
DOI: 10.1016/S0034-4877(98)80012-8
published in: Reports on Mathematical Physics
publication date: 1999-05-31T00:00:00Z
review text: The author's main idea is to use the nonholonomic bracket for the singular nonholonomic reduction. The motivation for the presented theory was born out of the observation that the obvious symmetry group in the tippe top does not act freely and that the exceptional orbits of the group action occur when the top is in one of the most interesting positions, the vertical ones. Of course, the central idea is to use invariants. The first example is the dynamic hatchet planimeter which exhibits asymptotic stability in the reduced two-dimensional problem that is expected to be Hamiltonian. Then, however, it can not have asymptotically stable orbits by a theorem of Hamiltonian mechanics. Singular reduction explains this seeming paradox and yields a geometrically faithful framework for studying singularity in the tippe top. Simply speaking, the idea is instead of looking at the distributional symplectic geometry of the problem to investigate the related `Poisson' bracket on the space of functions. This approach to the nonholonomic bracket brings a more direct way to see the relationship between the integrability of the constraints and the Jacobi identity for the bracket and a convenient computational framework for examinig the free particle by reinterpreting the Dirac construction for constrained brackets.
Mathematics Subject Classification ID: 37J60
Mathematics Subject Classification ID: 70H05
Mathematics Subject Classification ID: 70F25
Mathematics Subject Classification ID: 37K05
zbMATH DE Number: 1293242
zbMATH Keywords: nonholonomic bracket
zbMATH Keywords: nonholonomic reduction
zbMATH Keywords: group action
zbMATH Keywords: Hamiltonian mechanics
author: Larry Bates
MaRDI profile type: Publication
cites work: Proper group actions and symplectic stratified spaces
cites work: Nonholonomic reduction
cites work: A new analysis of the tippe top: Asymptotic states and Lyapunov stability
cites work: On the problem of steady motions stability of nonholonomic systems
cites work: Reduction of some classical non-holonomic systems with symmetry
cites work: Reduction of constrained mechanical systems and stability of relative equilibria
cites work: Analytical dynamics of discrete systems
cites work: Smooth functions invariant under the action of a compact Lie group
cites work: On the Hamiltonian formulation of nonholonomic mechanical systems
cites work: The local structure of Poisson manifolds
full work available at URL: s0034-4877(98)80012-8
OpenAlex ID: W1992780565
description: scientific article; zbMATH DE number 1293242