Geometric mechanics is a field with a very distinguished history. It has engaged a constellation of mathematicians and physicists including Hamilton, Lagrange, Newton, Euler, Poincare, Noether and Cartan — just to name a few. Yet, it can be difficult for students to get into this field because there seem to be so many prerequisites. A short list would include Lie groups and Lie algebras, ordinary differential equations, differential geometry, and the whole mechanism of calculus on manifolds. This book is one attempt to introduce the subject to undergraduates.

The name “geometric mechanics” itself is something of a misnomer because it suggests mechanics in the physicist’s sense of the kinematics and dynamics of moving particles, bodies or fluids. Yet the field is a good deal broader than that. For example, some of the basic ideas in geometric mechanics emerged in the principles of optics formulated by Galileo, Descartes, Huygens and Fermat. Of course, the dynamics of masses in motion, and especially celestial mechanics, do have a considerable role.

*Geometric Mechanics* surveys a modest piece of the field. It is divided into six primary sections. The first, “Fermat’s Ray Optics” introduces the main concepts and sets out the strategy for using Lie symmetry reduction that is used throughout the book. The key notion of a momentum map is also introduced here. The next section, “Newton, Lagrange, Hamilton,” surveys the contributions of that esteemed trio and compares Newtonian, Lagrangian and Hamiltonian formulations of mechanics. Then, in a section called “Differential Forms”, the author dives into the exterior calculus, symplectic manifolds and Lie derivatives.

The book’s three final sections deal with special topics. The first of these studies the resonance of two coupled nonlinear oscillators with an application to polarization states of travelling waves in nonlinear optical fibers. The second special topic is the elastic spherical pendulum, or swinging spring. Here the author applies the Lie symmetry reduction method to a system with a Lagrangian function averaged over rapid elastic oscillations. The last section takes up the Maxwell-Bloch equations that describe the interactions of laser and matter that give rise to so-called self-transparency.

This text was designed for third year undergraduates at Imperial College London. In the U.S., at least, this is reasonably challenging graduate-school-level material. The author’s writing is clear and well-organized, but he moves through a lot of topics very quickly. I learned the basics of geometric mechanics from Abraham and Marsden’s *Foundations of Mechanics*, but that, like the current book, is tough-going as an introduction. Students without a background in the area might be better served by Stephanie Singer’s Symmetry in Mechanics, a “gentle, modern introduction”, as she puts it.

The author has also written a companion volume, *Geometric Mechanics — Part II: Rotation, Translation and Rolling*, that takes up rolling round rigid bodies, tops, the special orthogonal and Euclidean groups and lots of related material.

Bill Satzer (wjsatzer@mmm.com) 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.