杏吧原创

Warwick

Celestial Encounters by Florin Diacu and Philip
Holmes, Princeton University Press, 拢19.95/$24.95, ISBN 0 691
02743 9

WHAT amazes me most about chaos theory is that it has held the attention of
the public for so long. So in the ten years it has been around, a market has
arisen for second-tier books that go into the subject more deeply, and hence
also more narrowly, than their predecessors. Florin Diacu used to be a
schoolteacher and is now at the University of Victoria in Canada. Philip Holmes,
currently at the University of Princeton, is one of the world leaders in
dynamical systems. Their topic is celestial mechanics, the study of bodies
moving in accordance with Newton鈥檚 law of gravity.

Diacu and Holmes concentrate on the history and mathematics of the n
-body problem, the motion of three or more bodies idealised as point masses. It
was here that chaos first came to the attention of mathematicians. The two-body
problem was partially solved by Isaac Newton and completed by Johann Bernoulli:
the bodies move in ellipses, parabolas or hyperbolas. The motion of three or
more bodies, however, is a vastly more difficult problem. The first person to
understand why was Henri Poincar茅, and the reason is that the motion can
be chaotic. It is in this setting that the phenomenon of chaos first became
apparent. The central theme of Celestial Encounters is the discovery of
chaos by way of celestial mechanics.

The book鈥檚 most attractive feature is a wealth of fascinating historical
material that illuminates the genesis of chaos. The key contributions of several
undeservedly obscure mathematicians are given due prominence. With luck, these
people will not be quite so obscure in the future. Chaos appears early enough to
capture our attention, and provides a fittingly up-to-the-minute climax.

A particular strength of Celestial Encounters is the impression,
conveyed with a deft touch, that mathematics is an international and collective
effort made by real people for real reasons in the real world. All of the
players in the drama, alive or dead, come over as human beings who happen to
have a passion for mathematics and an ability to fulfil that passion. This
feature alone justifies buying the book.

Poincar茅鈥檚 discovery of chaos came about when King Oscar II of Sweden
and Norway, on the advice of the Swedish mathematician G枚sta
Mittag-Leffler, announced a prize for research on any of four mathematical
problems, one being to solve the n-body problem in terms of convergent
series. Shortly before his death, Peter Lejeune-Dirichlet told Leopold Kronecker
that he had done just that, thereby proving that the Solar System is stable, but
he had not indicated his method. At the back of the prize jury鈥檚 mind was the
hope that Dirichlet鈥檚 alleged solution might be rediscovered.

Poincar茅 was at the height of his powers when he heard of the prize.
The monetary value was no more than moderate, but winning would have been of
immense value to his career. Over a period of three years he devised an entirely
new approach to mechanics, moving it away from quantitative analytic methods
towards qualitative geometric ones. He applied his qualitative methods to the
prize problem, and made sufficient inroads into it that on 21 January 1889 he
was awarded the prize. It is here that the detailed history becomes fascinating,
though convoluted: the truth has only recently become widely known, even among
mathematicians.

The manuscript that won the prize contained a serious error.

In mid-1887 Poincar茅 thought he had proved the Solar System鈥檚
stability鈥攑erhaps even reconstructed Dirichlet鈥檚 ideas. But no such
theorem appears in the printed version of his prizewinning memoir, published in
volume 13 of Acta Mathematica, a journal edited by Mittag-Leffler.
During 1889, when the memoir was being typeset and edited, Poincar茅 found
a major mistake. He telegraphed Mittag-Leffler on 30 November, asking him to
postpone publication until it could be rectified. This was eventually achieved,
though with difficulty, and the memoir duly appeared.

In 1985 the tale took a surprising turn, when Richard McGehee visited the
Mittag-Leffler Institute near Stockholm. In a box of uncatalogued papers,
McGehee found some copies of volume 13 of Acta Mathematica. But when he
compared one with the bound volume in the institute鈥檚 library, he discovered
that they were different. Mittag-Leffler had already printed copies, and
distributed many of them, before Poincar茅鈥檚 embarrassing error came to
light. Those copies were recalled, most of them were destroyed, and a
replacement volume was prepared. Poincar茅 paid the printing
bill鈥攚hich was probably more than the prize money.

What was the error? In a sensitive portion of the analysis, Poincar茅
had failed to consider one possible geometric configuration鈥攖he one that
subsequently led him to conclude that the three-body problem must have solutions
that we now call 鈥渃haotic鈥�. Such solutions are so complex that they have
elements of randomness, even though the mathematical equations are entirely
deterministic.

For Diacu and Holmes, that鈥檚 just chapter 1. In later chapters the authors
explore Stephen Smale鈥檚 creation of symbolic dynamics, which makes the
occurrence of chaos transparent. They introduce us to the work of Paul
Painlev茅 and his successors on singularities in the n-body
problem. Here, too, the reader will encounter the remarkable proofs of Zhihong
鈥淛eff鈥� Xia and Joseph Gerver that for five or more bodies there exist initial
configurations that disappear to infinity in a finite time. Diacu and Holmes
also include the epic proof by Xia that Arnold diffusion鈥攁n elusive kind
of slow chaos鈥攐ccurs in the three-body problem.

So much for the bouquets. Next, a brickbat. I don鈥檛 think that the attempt to
explain the mathematics at an advanced technical level succeeds in Celestial
Encounters. The level of sophistication that Diacu and Holmes require of
their readers is, I think, too high. And their trick of marking difficult
sections with an asterisk, with the meaning that 鈥測ou can omit this piece, or
skim it first time round鈥�, is an academic habit that does not work in a trade
book. If you have a story to tell, you should tell it and your readers should
read it. On the other hand, if certain passages can safely be omitted by the
reader, then they should be omitted beforehand by the book鈥檚 author. If these
passages are skimmable, the author should pre-skim them.

The practice adopted here leads to a turgid mini-course in dynamics occupying
pages 9 to 20. Just as the reader鈥檚 interest is turned on, along comes the big
turn-off or, for those who do skim, a worried feeling that the rest of the book
won鈥檛 make sense. Rightly so: a large chunk of the chapter on symbolic dynamics
is asterisked, but if you take the authors鈥� advice and skim it, the rest of the
chapter makes very little sense.

Recently overpublicised claims that chaos is just hype have resurfaced in the
media. Such claims have been around for a decade: they are often ill-informed
and typically a claimant rides on the same bandwagon that they are trying to
send careering into the ditch. Diacu and Holmes attempt to defuse such
criticisms by reducing the role of chaos to that of a bit-player in a
specialised drama.

I think that they are too defensive. The word 鈥渉ype鈥� trips easily off the
tongue but, it is worth thinking about what it means. Genuine hype occurs when
somebody with a lavish advertising budget spends it to promote a product.
Dynamical systems theorists ought to be able to distinguish this from a simple
case of positive feedback, in which a topic that spontaneously arouses media
interest becomes more interesting to the media because it has aroused media
interest. That鈥檚 not hype: it鈥檚 just what happens when science writers do their
job and present the excitement of science effectively.

We shouldn鈥檛 be apologising. We should be fighting back and explaining that
new areas of science and mathematics are exciting and important. And we should
not be too embarrassed by getting all that attention from the media, even if it
is sometimes overblown. After all nobody else is.