Symmetries and moment
type: Item
instance of: scholarly article
zbMATH Open document ID: 0982.70001
publication date: 2001-03-26T00:00:00Z
Mathematics Subject Classification ID: 70-02
Mathematics Subject Classification ID: 70H33
Mathematics Subject Classification ID: 70G45
Mathematics Subject Classification ID: 70E20
zbMATH DE Number: 1581514
zbMATH Keywords: equations of motion of rigid solid
zbMATH Keywords: principal bundle of periods of symplectic form
zbMATH Keywords: symplectic geometry
zbMATH Keywords: perturbation theory
zbMATH Keywords: d'Alembert principle
zbMATH Keywords: Euler-Lagrange equations
zbMATH Keywords: homogeneous formalism
zbMATH Keywords: Legendre transformation
zbMATH Keywords: symplectic manifold of geodesics
zbMATH Keywords: two-sphere
zbMATH Keywords: Poincaré disc
zbMATH Keywords: symmetry
zbMATH Keywords: momentum map
zbMATH Keywords: Noether theorem
zbMATH Keywords: Lagrange brackets
zbMATH Keywords: Poisson brackets
zbMATH Keywords: Darboux theorem
zbMATH Keywords: diffeological spaces
zbMATH Keywords: homotopy operator
zbMATH Keywords: de Rham complex
zbMATH Keywords: Stokes theorem
zbMATH Keywords: differentiable spaces
author: Patrick Iglesias-Zemmour
MaRDI profile type: Publication
title: Symmetries and moment
published in: Collection Enseignement des Sciences
review text: This book has its origin in a summer course given by the author between 1996 and 1997. It gives a clear and motivated introduction to symplectic mechanics, based on the notion of momentum mapping. The book is strongly inspired by the classical book by \textit{J.-M. Souriau} [Structure of dynamical systems. A symplectic view of physics. Transl. from the French by C. H. Cushman-de Vries. Progress in Mathematics (Boston, Mass.) 149 Boston: Birkhäuser. XXXIV (1997; Zbl 0884.70001)].NEWLINENEWLINENEWLINEThe book is divided into five chapters and six appendices. The first chapter describes the theory of perturbations introduced by Lagrange, as a motivation to study symplectic geometry. Indeed, perturbation theory has the germ of the notions of Lagrange (and Poisson) brackets and symplectic form. In the second chapter, the author gives a brief introduction to mechanics using the d'Alembert principle. A variational derivation is also given, and the Euler-Lagrange equations are obtained. The author uses a homogeneous formalism which consists in extending the Lagrangian one by adding a new variable. The Legendre transformation is studied in this context. The formalism is applied in chapter 3 to several examples: the computation of symplectic manifold of geodesics of the two-sphere \(S^2\) and on the Poincaré disc. The equations of motion of a rigid solid are also obtained.NEWLINENEWLINENEWLINEChapters 4 and 5 discuss the notions of symmetry and momentum map. In chapter 4 the author considers Lagrangians which are invariant by a Lie group, and proves the Noether theorem. In chapter 5 he studies in great detail the existence and properties of the momentum map. Several cohomologies are studied, and the principal bundle of periods of a symplectic form is constructed. As applications, the Bott-Thurston and Gelfand-Fuchs cocycles are recovered.NEWLINENEWLINENEWLINEThe book ends with six appendices. The first one discusses the fascinating history of Lagrange and Poisson brackets thr…
reviewed by: Manuel de León
description: scientific article