Geometric mechanics
type: Item
instance of: scholarly article
title: Geometric mechanics
zbMATH Open document ID: 1026.70001
published in: Lecture Notes in Mathematics
publication date: 2003-01-21T00:00:00Z
review text: This is an excellent monography on geometric mechanics which gives a geometric treatment of classical Newtonian mechanics, that is, mechanics in Riemannian manifolds. In fact, this is a topic where the bibliography is limited (see, for instance, the references quoted in this book), and thus this work contributes to alleviate this lack. The most standard geometric description of classical dynamical systems uses symplectic manifolds to describe both the Lagrangian and Hamiltonian formalisms of these systems. Nevertheless, Newtonian systems can also be described in an alternative but equivalent geometric way, taking into account that their configuration spaces can be considered as Riemannian manifolds. Then, using the covariant derivative induced by the corresponding Levi-Civita connection, the dynamical equations can be stated in terms of geodesics of the metrics. The book begins with the review of basic concepts of differential geometry (chapters 1 and 2), in particular (pseudo-) Riemanian manifolds (chapter 3). Then, after a brief classical presentation and discussion of Newtonian mechanics (chapter 4), the foundations of Riemannian mechanics are established (chapter 5). Next, the particular situations of systems with nonholonomic constraints and of vakonomic mechanics are treated in this framework (chapters 6 and 7). Finally, introductory remarks to the special and general relativity theories are given in chapters 8 and 9, using pseudo-Riemannian manifolds. Furthermore, some appendices are devoted to give insight into different subjects, namely: Lagrangian and Hamiltonian formalisms (appendix A), Möbius transformations and Lorentz group (appendix B), Maxwell formulation of Einstein's equations (appendix C), and viscosity solutions and Aubry-Mather theory (appendix D) (the last three are written in collaboration with other authors, Jose Natario and Diego Gomes).
Mathematics Subject Classification ID: 70-02
Mathematics Subject Classification ID: 70Hxx
Mathematics Subject Classification ID: 70G45
Mathematics Subject Classification ID: 70F25
zbMATH DE Number: 1856264
zbMATH Keywords: Newtonian mechanics
zbMATH Keywords: Riemannian manifolds
zbMATH Keywords: geometric mechanics
zbMATH Keywords: configuration spaces
zbMATH Keywords: covariant derivative
zbMATH Keywords: Levi-Civita connection
zbMATH Keywords: dynamical equations
zbMATH Keywords: geodesics
zbMATH Keywords: differential geometry
zbMATH Keywords: nonholonomic constraints
zbMATH Keywords: vakonomic mechanics
zbMATH Keywords: relativity
zbMATH Keywords: pseudo-Riemannian manifolds
zbMATH Keywords: Möbius transformations
zbMATH Keywords: Lorentz group
zbMATH Keywords: Einstein equations
zbMATH Keywords: viscosity solutions
zbMATH Keywords: Aubry-Mather theory
author: Waldyr Muniz Oliva
reviewed by: N. Román-Roy
MaRDI profile type: Publication
full work available at URL: b84214
OpenAlex ID: W4248894732
DOI: 10.1007/B84214
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description: scientific article; zbMATH DE number 1856264