Geometric Mechanics and Symmetry
From Finite to Infinite Dimensions
Series: Oxford Texts in Applied and Engineering Mathematics; 12;
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Product details:
- Publisher OUP Oxford
- Date of Publication 30 July 2009
- ISBN 9780199212903
- Binding Hardback
- No. of pages536 pages
- Size 241x162x34 mm
- Weight 922 g
- Language English
- Illustrations 32 line illustrations and 4 halftones 0
Categories
Short description:
A graduate level text based partly on lectures in geometry, mechanics, and symmetry given at Imperial College London, this book links traditional classical mechanics texts and advanced modern mathematical treatments of the subject.
MoreLong description:
Classical mechanics, one of the oldest branches of science, has undergone a long evolution, developing hand in hand with many areas of mathematics, including calculus, differential geometry, and the theory of Lie groups and Lie algebras. The modern formulations of Lagrangian and Hamiltonian mechanics, in the coordinate-free language of differential geometry, are elegant and general. They provide a unifying framework for many seemingly disparate physical systems, such as n–particle systems, rigid bodies, fluids and other continua, and electromagnetic and quantum systems.
Geometric Mechanics and Symmetry is a friendly and fast-paced introduction to the geometric approach to classical mechanics, suitable for a one- or two- semester course for beginning graduate students or advanced undergraduates. It fills a gap between traditional classical mechanics texts and advanced modern mathematical treatments of the subject. After a summary of the necessary elements of calculus on smooth manifolds and basic Lie group theory, the main body of the text considers how symmetry reduction of Hamilton's principle allows one to derive and analyze the Euler-Poincaré equations for dynamics on Lie groups.
Additional topics deal with rigid and pseudo-rigid bodies, the heavy top, shallow water waves, geophysical fluid dynamics and computational anatomy. The text ends with a discussion of the semidirect-product Euler-Poincaré reduction theorem for ideal fluid dynamics.
A variety of examples and figures illustrate the material, while the many exercises, both solved and unsolved, make the book a valuable class text.
The book provides a very comprehensive presentation of ideas and methods from geometric mechanics, aimed at the graduate-student level, but it could also be of interest for specialists who want to refresh their knowledge in this modern, elegant and unifying formulation of Lagangrian and Hamiltonian mechanics.
Table of Contents:
Preface
Acknowledgements
PART I
Lagrangian and Hamiltonian Mechanics
Manifolds
Geometry on Manifolds
Mechanics on Manifolds
Lie Groups and Lie Algebras
Group Actions, Symmetries and Reduction
Euler-Poincaré Reduction: Rigid body and heavy top
Momentum Maps
Lie-Poisson Reduction
Pseudo-Rigid Bodies
PART II
EPDiff
EPDiff Solution Behaviour
Integrability of EPDiff in 1D
EPDiff in n Dimensions
Computational Anatomy: contour matching using EPDiff
Computational Anatomy: Euler–Poincaré image matching
Continuum Equations with Advection
Euler–Poincaré Theorem for Geophysical Fluid Dynamics
Bibliography
