UK Nonlinear News Review

Geometric Mechanics by Darryl D Holm

Part I: Dynamics and Symmetry
Imperial College Press, 2008
ISBN-10: 1848161956, ISBN-13: 978-1848161955 (Hardcover: £43)
ISBN-10: 1848161964, ISBN-13: 978-1848161962 (Paperback: £17)

Part II: Rotating, Translating and Rolling
Imperial College Press, 2008
ISBN-10: 1848161557, ISBN-13: 978-1848161559 (Hardcover: £43)
ISBN-10: 1848161565, ISBN-13: 978-1848161566 (Paperback: £17)

Reviewed by Peter Hydon

For newcomers, the link between geometry and mechanics is not obvious. Mechanics deals with the motion of matter, whereas geometry is concerned with the properties of spaces and all that they contain. Geometric mechanics uses powerful tools and ideas from geometry to deal with the key features of a moving physical system, greatly simplifying the problem of understanding how the motion changes with time. This approach is immensely fruitful; its scope includes Hamiltonian systems, nonholonomic mechanics, fluid and solid mechanics, and control theory.

Lie, Klein and others introduced the idea that the geometric properties of a given space correspond to a Lie group of symmetries; that is, the properties are unchanged by a Lie group of transformations of the space. (Lie groups are groups that are also manifolds, so they are geometric objects in their own right.) For instance, in three-dimensional Euclidean space, lengths and angles are preserved by the Euclidean group, which is generated by translations, rotations and reflections. The special Euclidean group SE(3) is generated by translations and rotations only; it preserves all rigid bodies.

Now consider the free motion of a rigid body in three-dimensional Euclidean space. The body rotates about its centre of mass, which translates through space. So every motion of a rigid body can be regarded as a trajectory in the six-dimensional Lie group SE(3). In a comoving frame of reference, this reduces to a trajectory in the three-dimensional Lie group of rotations SO(3). The set of all possible trajectories is determined by the equation of motion, using one of two frameworks: Lagrangian (angular position - angular velocity) or Hamiltonian (angular position - angular momentum). At first sight, it seems that each of these are formulated in a six-dimensional space, but the equation of motion has symmetries: it is independent of the three components of angular position, so in each framework the problem is reduced to a set of differential equations on a three-dimensional space. Furthermore, the total angular momentum and kinetic energy are each preserved on every trajectory. This approach enables one to deal with the complete set of possible trajectories; for instance, one can establish the stability properties of each equilibrium.

The above example gives a small glimpse of the power that is obtained from the marriage of geometry and mechanics. However, applied mathematicians, engineers and physicists who are new to the subject commonly complain that it is hard to grasp. This is because of the need for a wide diversity of geometric ideas and structures, the use of some strange notation, and (until now) the lack of an undergraduate-level introduction that is geared to applications.

In these two books, Darryl Holm has provided just such an introduction. Indeed, almost every chapter in the first book focuses on a single substantial application. For instance, the eighty-three pages of the first chapter examine Fermat's ray optics from various points of view. It introduces many of the ideas that recur throughout both books, including the Lagrangian and Hamiltonian frameworks, Lie groups and Lie algebras, symmetry reduction in the Hamiltonian framework, and stability analysis. These ideas are carefully developed and extended throughout the first book, using examples of ever-increasing complexity as springboards for the introduction of new concepts. Reasonably straightforward exercises are scattered throughout the text; these are supplemented by an appendix containing fifty-two pages of 'enhanced courseworks' (some of which could be the basis for an undergraduate project).

After a short review of the key ideas from Part I, the second book focuses mainly on the Lagrangian framework. The pace of exposition is steady; applications are now used to illustrate ideas (rather than as a framework for a chapter, as in Part I). Quite sophisticated mathematics is used, but students who have worked through Part I carefully shouldn't have too much difficulty. Lagrangian symmetry reduction leads to the Euler-Poincare equation, whereas Hamiltonian reduction gives a Lie-Poisson equation. The climax of Part II is a clear explanation of the relationship between these two reductions, each of which provides useful insights. In particular, the Lagrangian framework enables one to deal with nonholonomic mechanics (rolling balls, for instance), with which the main part of the book concludes. Again, there is a reasonable number of simple and toughish problems.

Both books are very readable; the author has an easy, informal style. My first impression of Part I was of a joyously-written explosion of ideas and techniques from someone who wanted to give readers everything they need to know as quickly as possible. I liked the idea of looking at one application from different perspectives, but I think that some newcomers would find this a little overwhelming. However, everything's there, and with each successive reading it becomes easier to digest. Personally, I preferred the more measured pace of Part II, though it might feel less exciting. As for any work of this size, there are a few typographical errors. At one point (Part II, page 4), there is an incorrect implication that every Lie group is connected. But overall, the quality is very high.

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. At £17 each for the paperbacks, they are bargains!