Chapter 1
Introduction
In his 1954 plenary address to the International Congress of Mathematicians in Amsterdam, the Russian mathematician Andrey Kolmogorov announced a theorem that wowed the mathematical world. Mathematicians quickly realized that, if true as stated, the theorem resolved a paradox that had stood since Henri PoincarĂ©âs work at the end of the 19th century, and possibly also invalidated Ludwig Boltzmannâs ergodic hypothesis1 that lay at the foundations of statistical mechanics. Even more, if the theorem could be successfully applied to models of planetary motion based on Newtonian physics, the centuries-old goal of showing that the solar system is stable might finally be reached.
Itâs rare for a mathematical theorem to have such impact, and although Kolmogorov sketched a proof of the theorem that year (in a Russian mathematics journal [Kol54]), and discussed it a few years later (in the proceedings of the Amsterdam congress [Kol57]), the world still waited for definitive mathematical proof with all details spelled out. This came several years later in a series of remarkable papers by Kolmogorovâs young student Vladimir Arnold and the German-American mathematician JĂŒrgen Moser. Arnold was the first to show that Kolmogorovâs proof-techniques âworkedâ by using them to solve a previously intractable âcircle mapâ problem [Ar61]. The following year, Moser combined Kolmogorovâs proof-techniques with other methods to prove a specialized (low-dimensional) version of Kol-mogorovâs theorem [Mos62] (with one hypothesis that was unexpectedly weakâmaking the theorem unexpectedly strong). Then in 1963, Arnold proved a version of Kolmogorovâs theorem valid in all dimensions [Ar63a] (as Kolmogorov had announced in 1954), together with a closely related version that he applied to models of solar system stability [Ar63b], though under very restrictive conditions.
Thus was Kolmogorov-Arnold-Moser theory2 born, and it soon became customary to use the acronym KAM3 to refer to it. And although KAM theory has continued to evolve to the present day, passing through periods of fashionability and even mild controversy, it also unfortunately suffers undeserved obscurity among non-specialists.
1.1 What this book is, and how it came about
This book presents classical4 KAM theory in its broadest context. It is intended for mathematicians, physicists and other interested scientists whose training in classical mechanics stopped at the level of, say, (one of the editions of) H. Goldsteinâs book [Gold59], [Gold80], [GoldPS02] but who are nevertheless curious about what lies beyond. Experts may also find certain portions interesting, and I hope that they will add to or correct parts of the story with which theyâre especially familiar.5 Finally, the historical and speculative parts should also appeal to anyone interested in the history of ideas.
But let me be frank right from the start: this book will not teach you about KAM theory at a very deep mathematical level. I do not present a complete proof of a KAM theorem in these pages. Instead, the mathematical part of the story is connected by a century-long thread running from Henri PoincarĂ© to Kolmogorov, Arnold, Moser, and beyond. I trace this thread by way of a Hamiltonian function in modern notation, using it to show, in a simplified way, how mathematicians dealt with the problem of transforming a âslightly nonintegrableâ Hamiltonian into âintegrable form.â This approach should give the newcomer an idea of what the founders did, and a taste of the new techniques they (and others) created along the way. Since there is no shortage of rigorous mathematical treatments of KAM theory in the literature, readers who want to see complete proofs can choose from a wide selection.6
What does seem to be missing from the literatureâand what I provide here alongside the simplified mathematicsâis an overview of KAM theory, something that explains its content, history, and significance in relatively simple terms. I mean to clear up some common misunderstandings, to give a rough but understandable account of the main ideas, and to show how and why these ideas are important in mathematics, physics, and the history of science.
I can reveal one of the reasons for KAM theoryâs celebrity right away: Henri PoincarĂ© famously said that understanding perturbations of integrable Hamiltonian systems was the âfundamental problem of dynamics.â This innocuous sounding statement by the âfather of dynamical systemsâ conferred upon Hamiltonian perturbation theory (HPT) a fashionability that it enjoys to this day. Since KAM theory is the key result of HPT, it of course basks in the same glory; but it receives a furtherâvery dramaticâ boost from the fact that PoincarĂ© not only did not foresee KAM theory, but hinted that he thought it could not be true. In this sense (and in others to be explained) KAM theory went against the grain of its time.
This book grew out of an informal lecture on KAM theory that I gave in a number of places during the last decade. My view of the subject was formed by many years of being an American in Paris, where in the early days I worked in an area of HPT called Nekhoroshev theory7 which is closely related to KAM theory. Because Paris is a crossroads of European mathematics, I had a front-row seat from which to view many developments in the subject. As I looked on in amazement, over the years several odd things became evident. First of all, for a mathematical discipline, KAM theoryâor HPT generallyâis somewhat unsettled. Along each of several dimensions, thereâs a wider range of views than is ordinarily the case for a relatively mature mathematical subject. Let me run through just a few of these dimensions: physicists and mathematicians often differ markedly in their understanding of, use of, and enthusiasm for KAM theory. Researchers from different countries often seem to view and understand KAM theory differently. Occasionally, disagreement erupts over how much Kolmogorov proved in 1954 (some say his sketch-of-a-proof had such big gaps that it wasnât a proof at all; others say that it was complete enough to drop the A and M and simply call the KAM theorem âKolmogorovâs theoremâ). Still others think that C.L. Siegelâs name should be attached to the theorem (cf. §4.1 below to see why). In the early days after the announcement and proofs of KAM, there was some controversy over what the theorem might mean for mathematical physics, and physics generally. Did KAM really imply that the solar system was stable (or just that a âtoy modelâ of it was)? Did it really invalidate the ergodic hypothesis, thus throwing statistical mechanics into a foundational crisis? Later, in the area of HPT dealing with instability, a number of published results were found to have errors, and an uncharacteristic rancor and controversy erupted. Finally, although KAM theory sits right at the heart of chaos theory8 and is called by enthusiasts âone of the high points of 20th century mathematics,â there is remarkable ignorance of it among scientific journalists and chroniclers of chaos theory, especially in the U.S. All these thingsâand moreâare well known among experts, but experts themselves are rare.
1.2 Representative quotations and commentary
To show the reader that what I say above is not simply a way to generate interest in the subjectâthat KAM theory really does evoke a wide range of reactions among mathematicians and physicistsâI offer here some quotes from (relatively) recent books, in chronological order.
First, from an edition of the book most often used in American universities over the last half-century to teach classical mechanics to graduate students in physics, we have this (the only mention of KAM theory that appears9):
âOnly in the last few decades has the [solar system] stability question been freshly illuminated, by the application of new (and highly abstract) mathematical techniques. [. . .] A series of investigations, associated with the names C.L. Siegel, A.N. Kolmogorov, V.I. Arnold, and J. Moser, have shown that stable, bounded motion is possible for a system of n bodies interacting through gravitational forces only. [. . .] The brilliance of the achievement and the power of the new methods are probably of greater significance than the specific result, for the fate of the solar system will likely be determined by dissipative and other nongravitational forces.â
âH. Goldstein, Classical Mechanics (2nd Ed.), 1980 [Gold80] (p. 530)
Next, from a mathematics book that does include a chapter on KAM theory, with an outline of a proof:
âThe KAM theorem originated in a stroke of genius by Kolmogorov [. . . ]â
âP. Lochak & C. Meunier, Multiphase Averaging for Classical Systems, 1988 [LocM88] (p. 154)
From another mathematics book that provides careful and detailed treatments of many topics in perturbation theory comes a kind of apology for not treating KAM theory:
â. . . in the conservative case, the theory is very technical and deserves to be considered one of the high points of twentieth-century mathematics. It is called Kolmogorov-Arnold-Moser theory (frequently abbreviated to KAM), and is far too difficult to discuss in any detail here.â
âJ. Murdock, Perturbations. Theory and Methods, 1991 [Mur91] (p. 332)
One of the more interesting and revealing passages comes from a book intended for graduate students in physics:
âIn many ways the KAM theorem possesses sociological similarities to Gödelâs famous theorem in logic. (a) Both are widely known and talked about, yet many people are rather vague on what the theorems actually state, and very few have actually read the proofs, much less validated them. (b) Each has been called, by different mathematicians, the most important theorem of the twentieth century. (c) Neither is very useful for practical calculations: [...] the stable phase space estimated by the KAM theorem is typically too conservative to be of value.â
âL. Michelotti, Intermediate Classical Dynamics With Applications to Beam Physics, 1995 [Mic95] (pp. 305â306)
And finally, in a book by mathematicians popularizing the last century-and-a-half of achievements in dynamical systems and celestial mechanics, we have the following high praise:
â... the great edifice of KAM theoryâ
[And, at a later point in the book, also in reference to KAM theory:]
âone of the more remarkable mathematical achievements of this century . . .â
âF. Diacu & P. Holmes, Celestial Encounters, 1996 [DiH96] (p. 146, p. 165)
In these quotations, itâs interesting that authors seem compelled to pay tribute to KAM theory, to praise it and its inventors. Physicists (and even some mathematicians) seem also to want to avoid a direct encounter with it, saying itâs too âabstractâ or too âhard.â But in the quotations from the physics books by Goldstein and Michelotti, we also hear another reason for avoiding it: itâs not very useful. Once you know that many physicists think this, you realize that much of their praise is politely dismissive.
Mathematicians and physicists are generally civil with each other, and no one would write a strong statement about the uselessness of KAM theory in a book. But in spoken encounters over the years, Iâve heard much stronger statements and questions, such as âWhatâs so great about KAM theory?â10 or âWhat practical result has ever come from KAM theory?â, or even âIâm tired of hearing so much hype about KAM theory.â In this book, Iâll explore how ...