kam theory

December 1, 2013 17:6 World Scientific Book - 9in x 6in KAMstory

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 Poincares work at the end of the 19th century, and

possibly also invalidated Ludwig Boltzmanns 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.

Its rare for a mathematical theorem to have such impact, and although

Kolmogorov sketched a proof of the theorem that year (in a Russian math-

ematics journal [Kol54]), and discussed it a few years later (in the proceed-

ings of the Amsterdam congress [Kol57]), the world still waited for defini-

tive mathematical proof with all details spelled out. This came several

years later in a series of remarkable papers by Kolmogorovs young student

Vladimir Arnold and the German-American mathematician Jurgen Moser.

Arnold was the first to show that Kolmogorovs proof-techniques worked

by using them to solve a previously intractable circle map problem [Ar61].

The following year, Moser combined Kolmogorovs proof-techniques with

other methods to prove a specialized (low-dimensional) version of Kol-

mogorovs theorem [Mos62] (with one hypothesis that was unexpectedly

weakmaking the theorem unexpectedly strong). Then in 1963, Arnold

proved a version of Kolmogorovs theorem valid in all dimensions [Ar63a]

(as Kolmogorov had announced in 1954), together with a closely related

1Terms appearing in slanted text (as opposed to italics) are defined or discussed in the

glossary in Appendix F.

1


2 The KAM Story

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 in-

tended for mathematicians, physicists and other interested scientists whose

training in classical mechanics stopped at the level of, say, (one of the edi-

tions of) H. Goldsteins 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 theyre 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 mathemat-

ical part of the story is connected by a century-long thread running from

Henri Poincare to Kolmogorov, Arnold, Moser, and beyond. I trace this

thread by way of a Hamiltonian function in modern notation, using it to

2One hears KAM theory more often than the KAM theorem. As was evident right

from the start when the founders announced several different versions, there is no onetheorem, but instead many variations, each reflecting choices made in the underlying

hypotheses and methods of proof. Many of these variations will be detailed below. For

another succinct discussion of the early results focusing more on priority, see Part D.1.1of the readers guide in Appendix D.3The acronym KAM was coined in [IzC68] by F.M. Izrailev and B.V. Chirikov. Note

that in English, one customarily pronounces the three letters separately (K-A-M),whereas in Russian (and French), it is a true acronym, pronounced as the one-syllable

word kam.4By classical KAM theory, I mean the theory as it was originally developed for finite-

dimensional Hamiltonian systems and twist maps of the annulus. The expansion of KAM

theory outside its original framework is also touched upon in this book, but is not a main

emphasis.5See the books website http://thekamstory.wordpress.com/ to read or to submit

corrigenda.


Introduction 3

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 the-

ory 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 literatureand what I provide

here alongside the simplified mathematicsis 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 theorys celebrity right away:

Henri Poincare famously said that understanding perturbations of inte-

grable 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 furthervery dramatic

boost from the fact that Poincare 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

theoryor HPT generallyis somewhat unsettled. Along each of several

dimensions, theres a wider range of views than is ordinarily the case for a

6See Part D.1.2 of Appendix D for suggestions of where to find nice proofs of KAM

theorems.7So named after its developer Nikolai Nekhoroshev (19462008), a former student of

V.I. Arnold at Moscow State University. See 6.2 for more details.


4 The KAM Story

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

wasnt a proof at all; others say that it was complete enough to drop the

A and M and simply call the KAM theorem Kolmogorovs theorem). Still

others think that C.L. Siegels name should be attached to the theorem

(cf. 4.1 below to see why). In the early days after the announcementand 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 thingsand moreare 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 subjectthat KAM theory really does evoke a wide range of

reactions among mathematicians and physicistsI offer here some quotes

from (relatively) recent books, in chronological order.

First, from an edition of the book most often used in American uni-

versities 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):

8Its difficult to write the words chaos or chaos theory without quotation marks, asthese terms are quite elastic and have never been given universally accepted meanings by

mathematicians. (But see the chaos entry in the glossary in Appendix F.) This ambiguity

also makes them very useful terms, and I wont shy away from them in the sequel.9However, the latest (2002) edition of this book [GoldPS02] (now with coauthors) con-

tains a new chapter on classical chaos with a brief (2-page) section on KAM theory.


Introduction 5

Only in the last few decades has the [solar system]

stability question been freshly illuminated, by the applica-

tion 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 nongravita-

tional 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 apol-

ogy for not treating KAM theory:

. . . in the conservative case, the theory is very techni-

cal 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 sociolog-

ical similarities to Godels famous theorem in logic. (a)

Both are widely known and talked about, yet many people


6 The KAM Story

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 mathemati-

cians, the most important theorem of the twentieth cen-

tury. (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. 305306)

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 achieve-

ments of this century . . .

F. Diacu & P. Holmes, Celestial Encounters, 1996

[DiH96] (p. 146, p. 165)

In these quotations, its 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 its 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: its 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 the-

ory in a book. But in spoken encounters over the years, Ive heard much

stronger statements and questions, such as Whats so great about KAM

theory?10 or What practical result has ever come from KAM theory?,

or even Im tired of hearing so much hype about KAM theory. In this

book, Ill explore how remarks like these partly reflect a lack of knowledge,

and partly reflect justified frustration on physicists part.

Finally, following these brief quotations from books, I should also point

10This became the title of one of my talks to audiences of physicists.

January 9, 2014 15:39 World Scientific Book - 9in x 6in KAMstory

Introduction 7

out that in dynamical systems papers of the 1980s and 90s, it became so

common to see the term celebrated KAM theorem11 that you might think

the adjective celebrated had been permanently attached as part of the

theorems name.12

1.3 Remarks on the style and organization of this book

In these pages, Im going to tell KAM theory as a story, and Im going

to use several simplifying features to try to make the narrative more read-

able. First, at one end of a spectrum, I imagine people Ill call advocates

or enthusiasts for KAM theory (the reader can think of West European

or Russian mathematicians). At the other end, I imagine skeptics or de-

tractors of KAM theory (think of hard-boiled American physicists). Now

even if these are largely mythological characters, it will nevertheless be

more fun to think of things this way as we go. We can draw on a number

of stereotypes and cliches to keep us awake and make certain pointswe

already know that physicists and mathematicians like to tell jokes about

themselves and each other that turn on these cliches. Likewise, Americans

(and sometimes Britons) comfort themselves about their ignorance of conti-

nental European thought by picturing a fog of pointless theory emanating

from the old world, while Europeans are shocked by the crass pragmatism

in America, a place whose main contribution to philosophy has been to

enquire about the cash value of truth.13 From this point of view, my task

on the one hand is to try to educate the reluctant skeptics by penetrating

the fog of theory to find the underlying cash value of KAM theory. On the

other hand, if I occasionally make fun of a few enthusiastic theorists along

the way, so be it.

As mentioned earlier, I dont think KAM theory can be appreciated

properly without at least some knowledge of its history. In order to tell

it quickly and vividly, Ill recount many parts of the narrative using the

great man point of view, rather than the more painstaking process of

documenting all the individuals who contributed, though Ill at least list

11Whenever I see the word celebrated used this way, I cant help but think of MarkTwains short story The Celebrated Jumping Frog of Calaveras County. The reader

may recall that in that story, the said celebrated frog did not live up to its ownersexpectations, further emphasizing my feeling that much of the praise heaped upon KAM

theory is not wholly heartfelt.12I stopped looking once I found a dozen papers with this locution. UnfortunatelydareI admit it?one of the papers was my own.13As gleefully propounded by the Harvard-trained scholars C.S. Peirce and W. James.

January 9, 2014 15:39 World Scientific Book - 9in x 6in KAMstory

8 The KAM Story

some of them. Ill also employ some terminology anachronistically; a more

detailed approach would carefully follow changes in the meaning of termi-

nology over several centuries.14 In the places where the anachronisms are

especially misleading, Ill say so. Also regarding the way language is used

here, and despite the long time I spent in Europe and the European flavor

of the subject, Ive used something very close to ordinary American15 ver-

nacular to write this down. For my part, thats only natural, but its also

curiously unprecedentedKAM theory is rarely discussed in American.

Finally, this book comes with a lot of scholarly apparatus which the

reader may use according to his or her taste: in-text references to a single

long bibliography, several appendixes that together are almost as long as

the main text, and many footnotesa few pages have more footnotes than

ordinary text. I felt it was important to include the additional material as

a way of pointing to deeper layers of the story. As the reader will see, the

nature of KAM theory means that its story can be told on many levels.

14As Salomon Bochner puts it, [In history of science] more than in any other history,

the past discloses itself in the future. (See the interesting discussion from which thisfragment is quoted on p. 60 of [Boch66].)15Here American is used to mean the variety of English commonly spoken in the U.S.

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