UK Nonlinear News, March 2003
Springer, 2002. Hardback, xvii+571 pages. ISBN 0-387-95518-6.
Like Quantum Mechanics (QM), Geometric Mechanics (GM) is a basic way of thinking: GM is more than Newton's Law, just as QM is more than Schrodinger's equation. Once one masters either the QM, or the GM way of thinking, then one has a powerful framework for relating concepts and phenomena that organizes new information and unifies its presentation conceptually. GM and QM both have a quality of 'universality' about them that enables people to communicate rather freely among fields of endeavor in which either QM, or GM is the basic foundation. For example, the QM framework allows quantum chemists to talk freely with quantum information theorists and quantum condensed-matter physicists, because all have the same quantum intuition that developed during the inception of atomic physics. Likewise, GM allows the celestial mechanic to talk with the fluid dynamicist and control theorist, because all have the same geometric intuition about the Foundations of Mechanics available to them in the book of that title authored by Ralph Abraham and Jerry Marsden. Many developments have occurred in both QM and GM since these beginnings, and the intuition has expanded to account for these developments.
Indeed, when the papers collected in the present volume were presented at the Fields Institute in Toronto, about one hundred participants found themselves talking together about a multitude of topics, all in the same language of Geometric Mechanics.
The book's introduction reviews Jerry Marsden's contributions in many different scientific fields from a variety of perspectives. However, although the volume's papers come from a variety of directions, one should keep in mind the common language of GM. Consequently, readers familiar with the GM framework will find that they can easily read works in several different fields in this volume. Likewise, students wishing to learn this framework can find many examples of its applications. Students may also use the 'universal' aspects of GM to understand the richness of the examples in this volume in more depth than they might otherwise, in studying a text on a single field of application. Therefore, the relationships among the papers in this volume give it an interdisciplinary 'code breaking' aspect that is worth much more than merely the sum of its parts.
Any one of the articles in this volume can open the door to new research perspectives for a prepared mind. Conversely, the reader who recognizes the GM parallels between the various articles enriches his understanding of all of them. The GM framework is ideal for recognizing such parallels. In this framework, results from one field become informed by corresponding results in other fields. Thus, the articles in the book present perspectives in a variety of fields. However, the authors all write from within the same GM framework, so most readers will easily appreciate these parallels, once they know to look for them.
The Editors apparently charged the authors with the tasks of explaining and motivating their material carefully. Consequently, most of the papers are very accessible, with long introductions and good surveys of other contributions, in the typical 'Marsden style', as Ralph Abraham calls it in the Preface. The book's accessible style, combined with its unity in framework make it potentially useful as additional reading material for graduate students in applied mathematics and physics. Intructors might want to use this book to show students taking GM courses (taught, e. g., out of Marsden and Ratiu, Introduction to Mechanics and Symmetry) the breadth of the subject and also to give them hints for 'project papers'.
The volume contains research and survey articles describing applications of GM in several fields in which Jerry Marsden has also made significant contributions. These are cataloged at the end of the book as: Symplectic Reduction Theory; Lagrangian Reduction Theory; Fluid Mechanics and Plasma Physics; Relativistic Fields; Dynamical Systems in Mechanics; Nonlinear Elasticity; Control Theory; Variational integrators; and Dynamical Systems and Space Mission Design. The section titles in its table of contents provide another measure of the volume's broad scope: Elasticity and Analysis; Fluid Mechanics; Dynamical Systems; Geometric Mechanics; Geometric Control; and Relativity and Quantum Mechanics. The last section title also signifies the developing convergence between the GM and QM frameworks. This convergence is still 'under construction', although the parallels between these two frameworks have been a source of inspiration in mathematical physics research for several decades and will continue to be so in the future.
Finally, the volume is typeset so beautifully in a unified format, that it is satisfying to hold the book and pleasing to look at the articles, as well as to read them. This book should be in the collection of any serious student of GM. The volume is worth serious study, and it was well worth the long wait for the occasion of its production. Happy 60-th, Jerry. All the best.
UK Nonlinear News would like to thank Springer for providing a review copy of this book.
A listing of books reviewed in UK Nonlinear News is available.