Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions by Darryl D. HolmGeometric Mechanics and Symmetry: From Finite to Infinite Dimensions by Darryl D. Holm

Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions

byDarryl D. Holm, Tanya Schmah, Cristina Stoica

Paperback | August 15, 2009

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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 Hamiltonianmechanics, 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 andadvanced 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-Poincareequations 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-Poincare 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.
Darryl D. Holm spent thirty four years at Los Alamos National Laboratory before moving in 2005 to Imperial College London as Professor of Applied Mathematics. During his career, Darryl developed a wide range of applications of the geometric approach to dynamical systems. His main interest is in deriving and analyzing nonlinear evolutio...
Title:Geometric Mechanics and Symmetry: From Finite to Infinite DimensionsFormat:PaperbackDimensions:460 pages, 9.21 × 6.14 × 0 inPublished:August 15, 2009Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0199212910

ISBN - 13:9780199212910


Table of Contents

PrefaceAcknowledgementsPART I1. Lagrangian and Hamiltonian Mechanics2. Manifolds3. Geometry on Manifolds4. Mechanics on Manifolds5. Lie Groups and Lie Algebras6. Group Actions, Symmetries and Reduction7. Euler-Poincare Reduction: Rigid body and heavy top8. Momentum Maps9. Lie-Poisson Reduction10. Pseudo-Rigid BodiesPART II11. EPDiff12. EPDiff Solution Behaviour13. Integrability of EPDiff in 1D14. EPDiff in n Dimensions15. Computational Anatomy: contour matching using EPDiff16. Computational Anatomy: Euler; Poincare image matching17. Continuum Equations with Advection18. Eulerand#150;Poincare Theorem for Geophysical Fluid DynamicsBibliography