Order and Chaos in Dynamical Astronomy , George Contopoulos Springer-Verlag, New York, 2002. $79.95 (624 pp.). ISBN 3-540-43360-0
Imagine two black holes with just enough electric charge to cancel their mutual gravitational attraction. Then consider an uncharged test particle moving under the gravitational influence of those two fixed black holes and sharing a plane with them. In classical dynamics, the motion of the test particle turns out to be regular, but in general relativity it is chaotic.
Your reaction to Order and Chaos in Dynamical Astronomy will, I think, be much like your reaction to the two black holes. If you are charmed by the order-chaos dichotomy, you will find much to enjoy in the book. If it bothers you that nothing astronomers know of remotely resembles two fixed black holes, the book is probably not for you.
George Contopoulos has a halfcentury record of working on unusual but interesting problems. Best known for his work on high-order perturbation theory in stellar dynamics (the third integral), he also contributed—long before most others took any interest—to such now familiar areas in nonlinear dynamics as resonance overlap and period-doubling. It is interesting to have a pioneer’s view.
The book is not quite “my view of the field” though, more like “interesting topics I have worked on.” There is one clear theme: Completely regular and completely chaotic systems are both exceptional; most systems show a mixture of order and chaos. That is, it is typical to see neighboring trajectories diverge exponentially (the hallmark of chaos) in some parts of phase space but not in others. Moreover, the ordered and chaotic regions of phase space may be embedded within each other in incredibly complicated ways. Apart from that theme, however, the book does not have much in the way of general principles. The author tends to write in great detail about what most interests him, and then hurry through several other topics. In particular, astronomical problems other than in galactic dynamics make up only the last 5% of the book. On a smaller scale (the book is scale-invariant in this respect!), Lindstedt’s perturbation method is disposed of in two sentences at the end of the periodic-orbits section. Because of its style and structure, Order and Chaos in Dynamical Astronomy is unlikely to work as a textbook, but it makes a useful reference for researchers in and around the field.
The book’s biggest liability is probably the publisher. Springer has supplied a good cover and binding but not bothered with even basic copyediting or good-quality reproduction in figures, and the price tag will lose many potential readers.
Still, if you can afford it, there is plenty of interesting and useful information in the book. My favorite part was a discussion of the spectra of stretching numbers in chapter 2: I was always puzzled as to why Lyapunov exponents of chaotic orbits take so long to converge, and now I understand.
