UK Nonlinear News , February 1997.

## Reviewed by R.J. Martin

The objective of this book is to give an idea of some modern perspectives in the theory of integrable systems, and how to use them to obtain topological information, through examples coming from mechanics. The spinning top is the most celebrated example of a completely integrable system with two degrees of freedom.

Spinning tops have been investigated since the 18th century and from the pragmatist's point of view they provide the best justification for Lagrangian dynamics that one is likely to find at undergraduate level, simply because the Lagrangian analysis is so much more straightforward.

One finds on integrating the equations of the spinning top that the solution involves elliptic functions. These functions arise in several ways : (i) on integrating (1-k\sin^2\theta)^{-1/2}d\theta; (ii) as doubly-periodic functions; (iii) functions whose differential Galois group is an algebraic group variety of genus 1 (elliptic curve). Case (i) is the instance one meets in solving the equations directly; case (iii) is in the domain of the algebraic geometrists. One sees from this that the appearance of an elliptic integral is likely to provoke much algebraic interest.

The other strand is the Lax formulation of the system, that is, dX/dt=[X,Y]; as in general Y is allowed to depend on X, the equation is nonlinear. The system \dot{\Gamma}=\Gamma\cross\Omega, the Euler equations, is one such example.

This sets the scenery, but what I am not clear about is who is going to get any further. My experience of algebra extends beyond undergraduate level, but I still found the book too heavy-going. Several introductory chapters are required to make the book more accessible: one on the spinning top equations solved from first principles using Lagrangian dynamics, one on elliptic functions and their relation to algebraic curves, and one on Lax methods. This is not an exhaustive list.

The author describes the spinning top in what he calls a classical way'. But, as so often happens nowadays, the word classical' is used not when discussing a subject from first principles - and hence from a rudimentary point of view - but instead when discussing it using general, modern and sophisticated language. For the reader who knows something of the rudiments (and would like to be reminded of them), and little of the modern work, this is a cart-before-the-horse approach. Consequently the book is impenetrable unless you have done several postgraduate algebra courses, and its relevance to dynamics is very difficult to discern. On the other hand, maybe dynamicists are not the intended audience : but this is a shame because the subject matter is potentially of great interest to them.

Nine out of ten dynamicists will treat this as a book on algebra, and I suspect that the other one will too.

R.J. Martin, GEC Hirst Research Centre.
email: richard.martin@gecm.com.

UK Nonlinear News would like to thank Cambridge University Press for supplying a review copy of this book.

A listing of books reviewed in UK Nonlinear News is available at: http://www.amsta.leeds.ac.uk/Applied/news.dir/uknonl-books.html

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