Phase space: the fluid frontier

At the conclusion of my undergraduate course in Lagrangian dynamics in the 1960s, I was introduced to Liouville's theorem. I also studied fluid mechanics and the realisation that this theorem, which effectively said that the evolution of a class of dynamical systems in phase space was governed by the same arguments which, applied differently, lead to the continuity equation of fluid mechanics, was an insight into a new world. Since that time, advances in the theory of dynamical systems have been major and well reported.

Joseph McCauley's book aims to describe classical mechanics in this context.

In the opening chapter, the author states clearly that he will not "follow the mathematicians" by adopting an axiomatic or postulational approach, taking the view that this would fail to exhibit how the universal laws of motion were abstracted from established empirical laws. Then he gives a fascinating account of the history and foundations of mechanics - which would grace any mathematical text on the subject!

The second chapter gives a full account of the Lagrangian and Hamiltonian formulations of mechanics, showing clearly their relationship with the Newtonian formulation. Phase space is introduced in chapter three, together with the concept of a flow in phase space, and it is here that the first major theme, that of integrability, commences. Unfortunately, the introductory discussion of solvability versus integrability is not as transparent as one might wish.

A clearer statement of the definitions of the terms at this point would have been helpful, even if a full discussion had to be postponed. Bringing forward a little of the content of a later section (3.6) would aid clarity. I also found the conflicting statements concerning repellers on pages 98 and 99 rather disconcerting.

Chapters four and five deal with the important ideas of periodic motion and small oscillations about equilibria, and in the former the important distinction is made between periodicity and clockwork-like motion.

Opening with the quote by V. I. Arnold that "Hamiltonian mechanics is geometry in phase space", chapter six starts to deal in depth with integrable systems and to cover the groundwork for the rest of the text.

A simple example of a low-dimensional non-integrable system provides an introduction to the ideas of chaotic motion, and is followed by a full discussion of the phenomena.

The application of Lie groups to differential equations arising in mechanics has become increasingly important and chapter seven gives a good introduction to the concepts.

The author points out in the foreword that it was the realisation that every one-parameter transformation on two variables has an invariant, which lead to integrability being the main organising principle for the book.

Chapters eight and nine provide a fairly standard treatment of rotating frames and rigid body mechanics, with a return to Lagrangian dynamics in chapter ten, where issues of covariance and invariance are addressed prior to the establishment of a basis for configuration space.

I found chapter 11, which deals with gravity from both the Newtonian and the Einsteinian viewpoints and with the relation between invariance principles and the laws of nature, demanding but interesting reading.

Also worthy of note is the treatment in chapter 14 of damped-driven Newtonian systems. Here, period doubling, fractal orbits and strange attractors are all thoroughly and excitingly described.

The later chapters return to the theme of integrability versus non-integrability, and the book concludes by going back to its opening theme of mechanics and history, this time placing history in the context of mechanics!

I believe that this book is both a significant and timely, and indeed personal, contribution to the literature on mechanics, and one which will sit comfortably alongside other definitive Cambridge publications.

Nigel Steele is head of mathematics, School of Mathematical and Information Sciences, University of Coventry.