The KAM Story

A Friendly Introduction to the Content, History, and Significance of Classical Kolmogorov–Arnold–Moser Theory

H Scott Dumas

• Editorial: WSPC
• Año: 2014
• ISBN:
  9789814556606
• Categoría: Mathematics
• Categoría: Mathematics General

This is a semi-popular mathematics book aimed at a broad readership of mathematically literate scientists, especially mathematicians and physicists who are not experts in classical mechanics or KAM theory, and scientific-minded readers. Parts of the book should also appeal to less mathematically trained readers with an interest in the history or philosophy of science.

The scope of the book is broad: it not only describes KAM theory in some detail, but also presents its historical context (thus showing why it was a “breakthrough”). Also discussed are applications of KAM theory (especially to celestial mechanics and statistical mechanics) and the parts of mathematics and physics in which KAM theory resides (dynamical systems, classical mechanics, and Hamiltonian perturbation theory).

Although a number of sources on KAM theory are now available for experts, this book attempts to fill a long-standing gap at a more descriptive level. It stands out very clearly from existing publications on KAM theory because it leads the reader through an accessible account of the theory and places it in its proper context in mathematics, physics, and the history of science.

Contents:

• Introduction
• Minimum Mathematical Background
• Leading Up to KAM: A Sketch of the History
• KAM Theory
• KAM in Context: Questions, Consequences, Significance
• Other Results in Hamiltonian Perturbation Theory (HPT)
• Physical Applications
• Appendices:
  • Kolmogorov's 1954 Paper
  • Overview of Low-Dimensional Small Divisor Problems
  • East Meets West — Russians, Europeans, Americans
  • Guide to Further Reading
  • Selected Quotations
  • Glossary

Readership: Undergraduates, graduates, and researchers broadly interested in Hamiltonian perturbation theory, statistical mechanics, ergodic theory, Nekhoroshev theory, Arnold diffusion, nonlinear dynamics, dynamical systems, chaos theory, classical mechanics, and the history of these subjects.

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 Poincaré’s work at the end of the 19th century, and possibly also invalidated Ludwig Boltzmann’s 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.

It’s 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 Kolmogorov’s young student Vladimir Arnold and the German-American mathematician Jürgen Moser. Arnold was the first to show that Kolmogorov’s proof-techniques ‘worked’ by using them to solve a previously intractable ‘circle map’ problem [Ar61]. The following year, Moser combined Kolmogorov’s proof-techniques with other methods to prove a specialized (low-dimensional) version of Kol-mogorov’s theorem [Mos62] (with one hypothesis that was unexpectedly weak—making the theorem unexpectedly strong). Then in 1963, Arnold proved a version of Kolmogorov’s 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. Goldstein’s 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 they’re 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 Poincaré 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 ‘slightly nonintegrable’ Hamiltonian into ‘integrable form.’ 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 literature—and what I provide here alongside the simplified mathematics—is 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 theory’s celebrity right away: Henri Poincaré famously said that understanding perturbations of integrable Hamiltonian systems was the ‘fundamental problem of dynamics.’ This innocuous sounding statement by the ‘father of dynamical systems’ 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 further—very dramatic— boost from the fact that Poincaré 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 theory—or HPT generally—is somewhat unsettled. Along each of several dimensions, there’s 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 wasn’t a proof at all; others say that it was complete enough to drop the A and M and simply call the KAM theorem ‘Kolmogorov’s theorem’). Still others think that C.L. Siegel’s 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 ‘toy model’ 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 ‘one of the high points of 20th century mathematics,’ there is remarkable ignorance of it among scientific journalists and chroniclers of chaos theory, especially in the U.S. All these things—and more—are 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 subject—that KAM theory really does evoke a wide range of reactions among mathematicians and physicists—I 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):

“Only 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.”

—H. 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:

“The KAM theorem originated in a stroke of genius by Kolmogorov [. . . ]”

—P. 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:

“. . . 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.”

—J. 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:

“In many ways the KAM theorem possesses sociological similarities to Gödel’s 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.”

—L. Michelotti, Intermediate Classical Dynamics With Applications to Beam Physics, 1995 [Mic95] (pp. 305–306)

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:

“... the great edifice of KAM theory”

[And, at a later point in the book, also in reference to KAM theory:]

“one of the more remarkable mathematical achievements of this century . . .”

—F. Diacu & P. Holmes, Celestial Encounters, 1996 [DiH96] (p. 146, p. 165)

In these quotations, it’s 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 it’s too ‘abstract’ or too ‘hard.’ But in the quotations from the physics books by Goldstein and Michelotti, we also hear another reason for avoiding it: it’s 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, I’ve heard much stronger statements and questions, such as “What’s so great about KAM theory?”¹⁰ or “What practical result has ever come from KAM theory?”, or even “I’m tired of hearing so much hype about KAM theory.” In this book, I’ll explore how ...

Índice

• Cover
• Half Title
• Title Page
• Copyright
• Dedication
• Preface
• Acknowledgements
• Contents
• 1. Introduction
• 2. Minimum Mathematical Background
• 3. Leading Up to KAM: A Sketch of the History
• 4. KAM Theory
• 5. KAM in Context: Questions, Consequences, Significance
• 6. Other Results in Hamiltonian Perturbation Theory (HPT)
• 7. Physical Applications
• Appendix A. Kolmogorov’s 1954 paper
• Appendix B. Overview of Low-dimensional Small Divisor Problems
• Appendix C. East Meets West — Russians, Europeans, Americans
• Appendix D. Guide to Further Reading
• Appendix E. Selected Quotations
• Appendix F. Glossary
• Bibliography
• Index