Classical mechanics, arguably one of the oldest branches of science, is alive and well. Although it was once a standard part of a mathematical education, it has become a specialty area in mathematics while remaining a core field in physics. Classical mechanics grew up alongside mathematics. In the time of Newton its power was dramatically extended by the calculus. More recently differential geometry and the theory of Lie groups and Lie algebras have become important tools. The current book is intended to introduce the modern mathematical approach to classical mechanics (now often called “geometric mechanics”) to graduate students and advanced undergraduates. Inevitably, mathematicians call this approach “the right way to do mechanics”, but physicists have also come to value the insights that it offers.

In some ways this book seems to be two books in one. The first part concentrates on finite dimensional systems and proceeds at a fairly gentle pace. It would be accessible to a well-prepared undergraduate. The second takes up infinite dimensional systems; it is aimed at more advanced readers and concentrates on some recent applications such as computational anatomy and geophysical fluid dynamics. There is more than enough material in the book to support a two semester course. Exercises are generously interspersed throughout the book as are solutions to selected problems.

The key components of geometric mechanics are modern formulations of Lagrangian and Hamiltonian mechanics in the coordinate-free language of differential geometry. These provide a framework for many quite different physical systems, including n-body systems, rigid body motions and fluid mechanics, as well as electromagnetic and quantum systems. Symmetries of mechanical systems, typically represented by Lie group actions, are crucial. In part this is because here is a natural correspondence in mechanical systems between symmetry and conserved quantities: between translational symmetry and linear momentum, rotational symmetry and angular momentum, symmetry in time and energy, for example. Reduction by symmetry is a grand theme in mechanics; a symmetry together with the associated conserved quantity allows reduction of dimension of the phase space of a mechanical system.

This is an attractive text, suitable for self-study. The first part is intermediate in difficulty between Singer’s *Symmetry in Mechanics* and Marsden and Ratiu’s *Introduction to Mechanics and Symmetry.* The* *current book provides more details than are customarily offered, and that’s a real kindness to the reader. The second part has material that has not appeared in book form before, and is a good deal more challenging. The prerequisites include linear algebra, multivariable calculus, and knowledge of basic techniques for solving ordinary and partial differential equations.

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.