Geometric Mechanics - Part I: Dynamics And Symmetry (2Nd Edition) - Softcover

Holm, Darryl D.

Softcover

ISBN 10: 184816775X ISBN 13: 9781848167759

Publisher: ICP, 2011

Synopsis

See also GEOMETRIC MECHANICS -- Part II: Rotating, Translating and Rolling (2nd Edition) This textbook introduces the tools and language of modern geometric mechanics to advanced undergraduates and beginning graduate students in mathematics, physics and engineering. It treats the fundamental problems of dynamical systems from the viewpoint of Lie group symmetry in variational principles. The only prerequisites are linear algebra, calculus and some familiarity with Hamilton's principle and canonical Poisson brackets in classical mechanics at the beginning undergraduate level.The ideas and concepts of geometric mechanics are explained in the context of explicit examples. Through these examples, the student develops skills in performing computational manipulations, starting from Fermat's principle, working through the theory of differential forms on manifolds and transferring these ideas to the applications of reduction by symmetry to reveal Lie-Poisson Hamiltonian formulations and momentum maps in physical applications.The many Exercises and Worked Answers in the text enable the student to grasp the essential aspects of the subject. In addition, the modern language and application of differential forms is explained in the context of geometric mechanics, so that the importance of Lie derivatives and their flows is clear. All theorems are stated and proved explicitly.The organisation of the first edition has been preserved in the second edition. However, the substance of the text has been rewritten throughout to improve the flow and to enrich the development of the material. In particular, the role of Noether's theorem about the implications of Lie group symmetries for conservation laws of dynamical systems has been emphasised throughout, with many applications.

About this title

From the Inside Flap

This textbook introduces the tools and language of modern geometric mechanics to advanced undergraduates and beginning graduate students in mathematics, physics, and engineering. It treats the dynamics of ray optics, resonant oscillators and the elastic spherical pendulum from a unified geometric viewpoint, by formulating their solutions using reduction by Lie-group symmetries. The only prerequisites are linear algebra, calculus and some familiarity with Euler Lagrange variational principles and canonical Poisson brackets in classical mechanics at the beginning undergraduate level.

The ideas and concepts of geometric mechanics are explained in the context of explicit examples. Through these examples, the student develops skills in performing computational manipulations, starting from Fermat's principle, working through the theory of differential forms on manifolds and transferring these ideas to the applications of reduction by symmetry to reveal Lie Poisson Hamiltonian formulations and momentum maps in physical applications.

The many Exercises and Worked Answers aid the student to grasp the essential aspects of the subject. In addition, the modern language and application of differential forms is explained in the context of geometric mechanics, so that the importance of Lie derivatives and their flows is clear. All theorems are stated and proved explicitly.

The book's many worked exercises make it ideal for both classroom use and self study. In particular, a substantial appendix containing both introductory examples and enhanced coursework problems with worked answers is included to help the student develop proficiency in using the powerful methods of geometric mechanics.

About the Author

Professor Darryl Holm, received his PhD from University of Michigan in 1976, is a Chaired Professor in Mathematics at Imperial College London and a Laboratory Fellow in the Computation and Computer Science Division at Los Alamos National Laboratory. His work has ranged over the applications of Geometric Mechanics in a number of topics which remain of interest in his research. These topics include Fluid Dynamics, Emergent Singular Solutions of Nonlinear PDEs and Cardiomathematics.

Edition details

• Publisher: ICP
• Publication date: 2011
• ISBN 10: 184816775X
• ISBN 13: 9781848167759
• Binding: Paperback
• Language: English
• Edition number: 2
• Number of pages: 466