Recent results in the geometry of constrained systems
type: Item
instance of: scholarly article
title: Recent results in the geometry of constrained systems
zbMATH Open document ID: 1018.37041
DOI: 10.1016/S0034-4877(02)80025-8
author: Olga Rossi
published in: Reports on Mathematical Physics
publication date: 2003-01-22T00:00:00Z
review text: The paper presents a nice, comprehensive survey of the author's approach to the geometrical description of mechanical systems with nonholonomic constraints. This approach starts from the representation of an arbitrary system of second-order ordinary differential equations via a so-called dynamical form and its Lepage class on the second jet extension of a manifold fibred over a 1-dimensional base. The basic ingredients for such (unconstrained) dynamics are recalled in Section~2. Constraints are represented by a fibred submanifold \({\mathcal Q}\) of the first jet extension, leading to fundamental concepts such as the constraint ideal and the Chetaev bundle. An interesting result, quoted in Section~3, concerns the characterization of semi-holonomic constraints, for example by the property that the constraint ideal is a differential ideal. Semi-holonomic constraints turn out later (Section~6) to constitute the case in which the pullback of an unconstrained Lagrangian plays the role of Lagrangian for the constrained system. The main feature of this model is that constrained dynamical systems come from pulling back a representative of the Lepage class of a dynamical form to the constraint submanifold \({\mathcal Q}\). Constrained Lagrangian and Hamiltonian systems are discussed as particular cases within this general scheme.
Mathematics Subject Classification ID: 37J60
Mathematics Subject Classification ID: 70F25
Mathematics Subject Classification ID: 58E30
Mathematics Subject Classification ID: 70F17
Mathematics Subject Classification ID: 70H03
Mathematics Subject Classification ID: 70H45
zbMATH DE Number: 1857362
zbMATH Keywords: nonholonomic constraints
zbMATH Keywords: constrained Lagrangian systems
zbMATH Keywords: Lepage class
zbMATH Keywords: Hamiltonian systems
reviewed by: W. Sarlet
MaRDI profile type: Publication
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full work available at URL: s0034-4877(02)80025-8
OpenAlex ID: W2046101453
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description: scientific article; zbMATH DE number 1857362