| | Ivars Peterson's MathTrek |
November 15, 1999
The French mathematician Jules Henri Poincaré (1854-1912) made this observation in his 1908 essay "Science and Method." His route to that remarkable insight, which so aptly encapsulates a key feature of nonlinear dynamics (or what many people call, more loosely, "chaos" theory), was not a simple, direct path, however. Indeed, the triggering event was an embarrassing error, which precipitated a scandal of regal proportions.
In 1885, Magnus Gösta Mittag-Leffler (1846-1927), a mathematics professor at the fledgling school that eventually grew into the University of Stockholm, proposed a contest as part of the 60th-anniversary celebration of the birth of Oscar II, King of Sweden and Norway, which would occur on Jan. 21, 1889.
For the contest, mathematicians were invited to write an original paper addressing one of four questions. Proposed largely by Karl Weierstrass (1815-97), who had been Mittag-Leffler's teacher at the University of Berlin, these questions highlighted a number of key issues at the frontiers of mathematics research.
One of Weierstrass's questions concerned celestial mechanics and the stability of a collection of orbiting bodies: "Given a system of arbitrarily many mass points that attract each other according to Newton's laws, assuming that no two points ever collide, give the coordinates of the individual points for all time as the sum of a uniformly convergent series whose terms are made up of known functions."
The heavily promoted contest, with its prize of a gold medal and the modest sum of 2500 crowns, attracted wide attention and an array of entries. Among these was a hefty contribution from Poincaré, who only a few years earlier, at the age of 27, had become a professor at the prestigious University of Paris.
Equipped with an incredibly acute memory, Poincaré worked intuitively, often figuring out problems in his head as he restlessly paced back and forth in his quarters. Only after he had worked out an idea in his mind would he commit it to paper, and his written work typically showed signs of hasty composition.
Indeed, Poincaré intensely disliked retracing his steps to fill in gaps and to tidy up his reasoning and mathematical language. Satisfied that he had surmounted the crucial barriers and hacked out a rough path to the solution, he would plunge headlong into the next daunting mathematical thicket.
Poincaré's idiosyncratic approach to mathematics had been evident from the start. In 1875, at the age of 21, he had entered the School of Mines planning to become an engineer, but he spent nearly all his spare time developing a novel approach to differential equations. To do this, he advanced the study of differential equations from numbers, formulas, and the manipulation of algebraic equations to geometry, curves, and the visualization of flows.
Henri Poincaré
These investigations proved so fruitful that three years later he was able to present his results as a doctoral thesis at the University of Paris. Gaston Darboux (1842-1917), a member of Poincaré's thesis committee, was later to recall: "At first glance, it seemed clear to me that [Poincaré's thesis] was out of the ordinary and fully deserved to be accepted. Certainly it contained enough results to furnish material for several good theses."
"But," he continued, "it must be said without hesitation if an accurate idea is to be given of the way in which Poincaré worked, many points required corrections and explanations. . . . He willingly did the corrections and tidying which seemed necessary to me. But when I asked him to do so, he explained to me that he had many other ideas in his head; he was already occupied with some of the great problems whose solution he was to give us."
The promising geometric perspective that Poincaré had pioneered for understanding differential equations offered a novel approach for solving problems in celestial mechanics, so he entered Mittag-Leffler's contest. It was a chance to test his methods on the global behavior of the solutions to the differential equations of celestial mechanics.
Faced with the extreme difficulty of properly interpreting the interminable series that typically arise in solving those equations (the route suggested by the contest problem), Poincaré shifted the problem from specific cases to solutions in their totality. Inventive and revolutionary, his contest entry explained and then used geometric ideas to characterize the motion of three gravitationally interacting bodies (see Lunar Shadows, August 23, 1999).
Poincaré's complex, lengthy paper, however, brought the contest's judges, especially Weierstrass, into unfamiliar mathematical territory. They needed time to sort through Poincaré's complicated reasoning, but they were also under great pressure to complete their deliberations in time for the celebrations marking King Oscar II's birthday.
Finally, the pressure to make an announcement overruled whatever reservations the ailing, weary Weierstrass may have had. Poincaré's paper was clearly the best submission and deserved the prize.
Weierstrass reported to Mittag-Leffler, "You may tell your sovereign that this work cannot indeed be considered as furnishing the complete solution of the question proposed, but that it is nevertheless of such importance that its publication will inaugurate a new era in the history of celestial mechanics."
Poincaré was declared the winner. In January, 1889, Poincaré came to Sweden to accept his prize, in an atmosphere of considerable pomp and ceremony.
The next step was publication of Poincaré's award-winning paper in Mittag-Leffler's journal Acta Mathematica. The long process of editing, typesetting, and printing took place from July to November, 1889. During that prepublication period, one of the journal editors raised questions about certain passages in Poincaré's manuscript.
In the course of responding to the questions, Poincaré realized that he had made a serious mistake in another section of the paper. In the three-body problem, Poincaré believed that he had proved certain stability results. Those results, however, did not hold up under further examination.
On November 30, Poincaré telegraphed Mittag-Leffler to stop the printing, pending the arrival of a letter explaining the error. Printing of the journal with Poincaré's article had already been completed, however. Preliminary copies were in circulation among a few prominent mathematicians and astronomers.
This was a political disaster for Mittag-Leffler. He took the drastic measure of retrieving all the distributed copies of the journal with Poincaré's original article and had them destroyed, along with (nearly) all other copies in hand.
Mittag-Leffler also persuaded Poincaré to revise the proof and submit a new paper, which would be published in place of the original. Poincaré spent several frantic months rethinking everything he had done.
It was a massive revision. The new paper was nearly 100 pages longer than the original. In the revised introduction, Poincaré thanked Lars Edvard Phragmén (1863-1937), a colleague of Mittag-Leffler. "It is he, who in calling my attention to a delicate point, made it possible for me to discover and correct an important error," Poincaré wrote. He did not mention the nature of his error, however.
Many things in the revised paper were not changed from the original. In the revised sections, however, Poincaré provided the first indications of the true complexity of Newtonian dynamics, even in a system as apparently simple as three gravitating bodies. Indeed, he gave the first example of the sensitive dependence on initial conditions characteristic of chaotic behavior.
In his initial effort, Poincaré had failed to consider one possible geometric configuration--just the one that eventually led him to conclude that the three-body problem must have solutions that we would now describe as "chaotic."
That's something we can readily visualize and experiment with on a computer today. Poincaré came to that insight without the benefit of computers or extensive calculation. In rethinking his work, he confronted, then accepted the possibility that there is room for the wildly unpredictable in deterministic systems.
Interestingly, nothing about the three-body equations themselves had changed. Instead, using his new way of "viewing" differential equations, Poincaré could see something that was built into those equations all along but had previously escaped notice.
It was a bold, daring step to take, one made under tremendous pressure and in extremely trying circumstances. Making that leap was a real tribute to Poincaré's genius.
Copyright 1999 by Ivars Peterson
References:
Barrow-Green, J. 1997. Poincaré and the Three Body Problem. Providence, R.I.: American Mathematical Society.
Diacu, F. 1996. The solution of the n-body problem. Mathematical Intelligencer 18(No. 3):66.
Diacu, F., and P. Holmes. 1996. Celestial Encounters: The Origins of Chaos and Stability. Princeton, N.J.: Princeton University Press.
Gutzwiller, M.C. 1998. Moon-Earth-Sun: The oldest three-body problem. Reviews of Modern Physics. 70(April):589.
Holmes, P. 1990. Poincaré, celestial mechanics, dynamical-systems theory and "chaos." Physics Reports 193(September):137.
Jackson, A. 1999. The dream of a Swedish mathematician: The Mittag-Leffler Institute. Notices of the American Mathematical Society 46(October):1050.
Peterson, I. 1993. Newton's Clock: Chaos in the Solar System. New York: W.H. Freeman.
PoincarŽ, H.J. 1993. New Methods of Celestial Mechanics, edited and introduced by D. Goroff. New York: American Institute of Physics.
______. 1890. Sur le problème des trois corps et les équations de la dynamique. Acta Mathematica 13:1.
The Internet Encyclopedia of Philosophy features a biography of Henri Poincaré at http://www.utm.edu/research/iep/p/poincare.htm. A brief biography also appears at http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Poincare.html.
Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.