Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions

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Darryl D. Holm (Professor, Mathematics Department, Imperial College London) Tanya Schmah (Department of Computer Science, University of Toronto and Department of Mathematics, Macquarie University, Australia)

Geometric Mechanics and Symmetry

From Finite to Infinite Dimensions

Darryl D. Holm (Professor, Mathematics Department, Imperial College London), Tanya Schmah (Department of Computer Science, University of Toronto and Department of Mathematics, Macquarie University, Australia), Cristina Stoica (Department of Mathematics, Wilfrid Laurier University, Canada)

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English

Oxford University Press
15 July 2009

Mathematics & Sciences; Calculus; Geometry; Applied mathematics; Fluid mechanics

Series: Oxford Texts in Applied and Engineering Mathematics

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 nDSparticle 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.

By: Darryl D. Holm (Professor Mathematics Department Imperial College London), Tanya Schmah (Department of Computer Science, University of Toronto and Department of Mathematics, Macquarie University, Australia), Cristina Stoica (Department of Mathematics, Wilfrid Laurier University, Canada)
Imprint: Oxford University Press
Country of Publication: United Kingdom
Volume: 12
Dimensions: Height: 233mm, Width: 155mm, Spine: 30mm
Weight: 791g
ISBN: 9780199212910
ISBN 10: 0199212910
Series: Oxford Texts in Applied and Engineering Mathematics
Pages: 526
Publication Date: 15 July 2009
Audience: College/higher education , A / AS level
Format: Paperback
Publisher's Status: Active

Preface Acknowledgements PART I 1: Lagrangian and Hamiltonian Mechanics 2: Manifolds 3: Geometry on Manifolds 4: Mechanics on Manifolds 5: Lie Groups and Lie Algebras 6: Group Actions, Symmetries and Reduction 7: Euler-Poincaré Reduction: Rigid body and heavy top 8: Momentum Maps 9: Lie-Poisson Reduction 10: Pseudo-Rigid Bodies PART II 11: EPDiff 12: EPDiff Solution Behaviour 13: Integrability of EPDiff in 1D 14: EPDiff in n Dimensions 15: Computational Anatomy: contour matching using EPDiff 16: Computational Anatomy: EulerDSPoincaré image matching 17: Continuum Equations with Advection 18: EulerDSPoincaré Theorem for Geophysical Fluid Dynamics Bibliography

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 evolution equations for multiscale phenomena. Applications of these equations have ranged from nonlinear optical pulses used in telecommunications, to turbulence modeling for global ocean circulation and climate prediction, to template matching for the shapes of biomedical images, to directed self-assembly in nanoscience. The solution behavior of these equations includes solitons (governed by the Camassa-Holm equation), vortices and turbulence (modelled by the LANS-alpha equation) and emergent singularities (modelled by the EPDiff equation) representing the sharp edges that appear in biomedical images. Tanya Schmah completed her PhD in mathematics in 2001 at the Swiss Federal Institute of Technology in Lausanne. She has held lectureships at the University of Warwick (U.K.) and Macquarie University (Australia), and is currently working in the Department of Computer Science at the University of Toronto. She has a wide range of interests in mathematics and computer science, including symmetric Hamiltonian systems and machine learning. Cristina Stoica has a Diploma in Mathematics-Mechanics from the University of Bucharest (1991) and possesses a Doctor of Sciences degree in Astronomy awarded by the Astronomical Institute of the Romanian Academy (1997). She also holds a PhD in Applied Mathematics from the University of Victoria, Canada (2000). Currently she is a faculty member at Wilfrid Laurier University, Canada. Her main interests lie at the intersection of dynamical systems and mathematical physics.

Reviews for Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions

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. Jean-Francois Ganghoffer, Journal of Geometry and Symmetry in Physics ...the book makes much more than merely allowing students to pass from one book to the other, and turns out to be a very well written and self-contained treatment of the interplay between mechanics and symmetry. European Mathematics Society Throughout the text the exposition is very clear, and this is in big part due to the extensive use of detailed examples, which is probably one of the strongest pedagogical points of this text when compared with other text books with similar subjects and targets...This makes this text the starting point for any researcher interested in getting started in the field of geometric models for continuum systems. Journal of Geometric Mechanics