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Symmetry in Mechanics: A Gentle, Modern Introduction

Stephanie Frank Singer
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
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Symmetry in Mechanics is an introduction to modern symplectic geometry and classical mechanics.  It does not attempt to be a comprehensive introduction to either subject.  Instead, as the subtitle — "A Gentle, Modern Introduction" — suggests, this is an accessible text on the modern mathematical approach to mechanics, an alternative to the far more challenging Foundations of Mechanics of Abraham and Marsden or Arnold's Mathematical Methods of Classical Mechanics, for example.

The book is aimed at undergraduates in mathematics or physics.  It's for real students, as the author says, "talented but appreciating review and reinforcement of past course work; willing to work hard, but demanding context and motivation for the mathematics they are learning."  The prerequisites are multivariable calculus and basic linear algebra.

The story begins and ends with a derivation of Kepler's laws for the motion of two bodies interacting under Newtonian gravitation.  The first time through, Kepler's laws are derived in a traditional ("physics style") manner. One begins with Newton's laws in twelve-dimensional phase space, changes variables a few times, takes advantage of the conservation of linear momentum, energy and angular momentum and eventually reduces to a set of explicitly solvable ordinary differential equations.  The reader is meant to see the ad hoc flavor of this approach.

The last chapter revisits the Kepler problem, this time with the machinery of symplectic geometry.  Applications of symmetry are direct with symplectic reduction, and one can avoid writing down equations in coordinate form until the very end.  In two stages of reduction, one goes from the full twelve-dimensional phase space to a two-dimensional reduced phase space, after which one can recover Kepler's laws directly.

Between the first and last chapters, the author builds the tools of symplectic geometry as well as the necessary background in mechanics.  Instead of creating a general theory, the author keeps the development as simple and coherent as possible.  Symplectic manifolds are introduced as the natural phase spaces of mechanical systems; differential forms, vector fields on manifolds, and Hamiltonian energy functions are also presented in this context.  Lie group actions are introduced as a natural way to represent symmetries, and Lie algebras play a similar role for infinitesimal symmetries.  Finally, the concept of a momentum map in conjunction with group actions clarifies the relationship between symmetries (such as rotational symmetry) and conserved quantities (such as angular momentum).

This attractive text makes serious efforts to address real students and their potential difficulties.  There are more than one hundred exercises of varying difficulty integrated with the text, and solutions of many of these are provided in an appendix.  The author is comfortable with both the mathematician's and the physicist's views of this subject, and she reminds her readers of the value of both perspectives.

The exposition is uneven in spots.  Some topics are over-explained, while other more difficult subjects are treated rather quickly.  The early section on Kepler's laws doesn't make it clear that the author is considering only the bounded motion case.  Overall, however, this is a carefully thought out and well-executed textbook.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

 Preface * Preliminaries * 1. The Two-Body Problem * 2. Phase Spaces are Symplectic Manifolds * 3. Differential Geometry (Optional) * 4. Total Energy Functions are Hamiltonian Functions * 5. Symmetries are Lie Group Actions * 6. Infinitesimal Symmetries are Lie Algebras * 7. Conserved Quantities are Momentum Maps * 8. Reduction and the Two-Body Problem * Solutions to Exercises * Bibliography * Index