*Celestial Encounters*is a historical study of this, looking at dynamical systems theory going back to Poincaré and taking a qualitative and geometrical approach.

There are almost no equations in *Celestial Encounters*, but that is
because of the way the mathematics is presented, not because it has
been trivialised.

"Liapunov's doctoral thesis is the starting point of the modern theory of stability. The definition he gave of this fundamental notion is simple and powerful. Consider a differential equation and a particular solution in phase space, such as the one represented in figure 4.3 by the curve C. We say that this solution isstableif all other solutions of the same equation starting close to C remain close to it for all time. In other words, there is a (narrow) strip in the phase space, containing C, such thateverysolution curve having a point in the strip will remain forever inside that strip. This greatly generalizes the notion of stability of a fixed point or equilibrium solution, for Liapunov's definition applies toanykind of solution: steady, periodic, or more exotic ones."

About a sixth of the material is in subsections marked as more technical, which can be skipped (though the resumption after the first such excursion immediately uses the term "homoclinic tangle", which will mean nothing to someone who has skipped it).

*Celestial Encounters* is dominated by biographical material, but this
remains focused on the scientific story, giving a feel for how ideas have
been discovered, shared, and cross-fertilised within the mathematical
community. It is notable that both authors are active researchers in
the field rather than, as with many popular science books, outsiders.

An unusual feature is that some of the life stories are lightly fictionalised, with dramatic recreation of "scenes and conversations for which there are no direct data". This is sometimes odd, but it is always obvious when it is being done, as the following example shows.

"Jürgen Moser read through the proof again and again, trying to understand all the unwritten details. He was wondering why they were not more clearly stated in the article, since he did not find them at all obvious. An editor from theMathematical Reviews, a journal that provides a useful service by printing brief summaries of technical articles, had asked him to produce a short account of a paper. Moser was finding it hard to complete this task, which under normal circumstances is a fairly routine matter. He would have preferred to avoid the struggle that was going on within himself, but, once started, he knew he would have to continue to the end."

Each of the five chapters in *Celestial Encounters* follows a separate
chronological thread, focusing on a few key ideas and protagonists.

The first chapter introduces the n-body problem and concepts of phase spaces, manifolds, and local and global existence. It is centred on Henri Poincaré and tells the story of the prize awarded by King Oscar II of Sweden; his winning this was complicated by the realisation of an error in his work, associated with chaotic behaviour, which forced its republication. Some more technical material here introduces fixed points, first returns, and homoclinic tangles.

Chapter two turns to symbolic dynamics. The protagonists here include George Birkhoff, Stephen Smale, and Russian mathematicians K. Sitnikov and V.M. Alekseev; there is also some discussion of the Cold War relationship between Soviet and Western mathematics. Topics covered include the fixed-point theorem, horseshoe maps, the shift map, and whether "chaos theory" is a new science or a collection of methods in mathematics.

Presenting work on collisions and other singularities of the n-body problem, chapter three follows Paul Painlevé, Hugo von Zeipel, Donald Saari, Richard McGehee, Joseph Gerver, and Zhihong Xia. This touches on attempts to construct systems with non-collision singularities, fractals and Lebesgue measure, regularization of collisions, and collision manifolds,

Looking at stability, key figures covered are Laplace (Pierre Simon), Lagrange (Joseph Louis), Siméon Denis Poisson, Romanian mathematician and educationalist Spiru Haretu (presumably included because Diacu is of Romanian origin), and Aleksandr Liapunov. The story here features the question of the long-term stability of the solar system, Laplace's famous "no need of that hypothesis" retort to Napoleon, and qualitative approaches to different kinds of stability.

The final chapter turns to KAM theory, the result of work on perturbations of quasi-periodic orbits by (among others) Andrei Kolomogorov, Vladimir Arnold and Jürgen Moser. Most of this chapter is marked as "technically difficult" and it involves Hamiltonians, invariant tori, twist maps, and Arnold diffusion.

Given the relative complexity of the material it covers, the most obvious
audience for *Celestial Encounters* will be those with a mathematical
background, perhaps students doing a course on dynamical systems who
want some historical context. It is possible to skip the more technical
material, however, and Diacu and Holmes do a good job of providing
motivation and context, so it should also appeal to lay readers who
don't want to tackle a textbook but are after a bit more substance than
popular books on "chaos" offer.

November 2010

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