# GEOMETRIC MECHANICS: Part 2, Rotating, Translating and Rolling (Pt. II)

## Darryl D. Holm

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This textbook introduces the tools and language of modern geometric mechanics to advanced undergraduate and beginning graduate students in mathematics, physics, and engineering. It treats the dynamics of rotating, spinning and rolling rigid bodies from a geometric viewpoint, by formulating their solutions as coadjoint motions generated by Lie groups. The only prerequisites are linear algebra, multivariable calculus and some familiarity with Euler Lagrange variational principles and canonical Poisson brackets in classical mechanics at the beginning undergraduate level.

Variational calculus on tangent spaces of Lie groups is explained in the context of familiar concrete examples. Through these examples, the student develops skills in performing computational manipulations, starting from vectors and matrices, working through the theory of quaternions to understand rotations, and then transferring these skills to the computation of more abstract adjoint and coadjoint motions, Lie Poisson Hamiltonian formulations, momentum maps and finally dynamics with nonholonomic constraints.

The 120 Exercises and 55 Worked Answers help the student to grasp the essential aspects of the subject, and to develop proficiency in using the powerful methods of geometric mechanics. In addition, all theorems are stated and proved explicitly. The book s many examples and worked exercises make it ideal for both classroom use and self-study.

Contents: Galileo; Newton, Lagrange, Hamilton; Quaternions; Quaternionic Conjugacy; Special Orthogonal Group; The Special Euclidean Group; Geometric Mechanics on SE(3); Heavy Top Equations; The Euler Poincaré Theorem; Lie Poisson Hamiltonian Form; Momentum Maps; Round Rolling Rigid Bodies.

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Review:

Both books are very readable; the author has an easy, informal style ... These two books, written by one of the masters in geometric mechanics, provide an accessible way into the subject for newcomers; they also give a unique perspective for those who are not so new. --Professor Peter Hydon, UK Nonlinear News Review

It is a continuation of the elegant presentation of geometric mechanics of Volume 1. --Quarterly of Applied Mathematics

Students who work carefully through the material in these volumes will amass a formidable armory of mathematical techniques and will be well equipped to attack new and challenging problems in mechanics ... The appendices contain a collection of valuable example problems that are suitable for both homework and enhanced coursework. There are also numerous exercises scattered throughout the text to allow readers to evaluate their progress. --SIAM Review

In both Parts, physical examples play an important role. There are also excellent references to recent literature as well as nice historic contexts ... the Geometric Mechanics books are valuable additions to the literature on geometric mechanics and symmetry. The books are well written and pleasant to read. I can recommend anyone teaching a course on applied geometric mechanics to consider one or both books for text books or recommended reading. The books would be a good starting point for anyone interested in learning more about applied geometric mechanics and symmetry. The many references to recent literature put the reader in a good position to find out about the latest research in this areaw. --Journal of Geometric Mechanics

This two-volume book fills a niche in the geometric mechanics literature at the crossroads between mathematics and physics/engineering and between elementary and advanced texts on theoretical mechanics. --Mathematical Reviews

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