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The KAM Story

H. Scott Dumas
World Scientific
Publication Date: 
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
William J. Satzer
, on

The KAM Story is a tale told about a theory that was built through the loose collaboration of Andrey Kolmogorov, Vladimir Arnold, and Jürgen Moser. KAM theory incorporates a collection of theorems and an amalgam of related approaches to problems in classical mechanics and particularly celestial mechanics. The author says that his intention is to present classical KAM theory (in a broad sense) at a level accessible to mathematicians, physicists, and others whose background includes something equivalent to Goldstein’s Classical Mechanics. But the book is really more accessible than that — as long as the reader is willing to do some judicious skipping around.

This is mathematical story-telling using simplified mathematics to develop an overview of KAM theory that explains its history, significance, and content. Proofs of results can be found in many places (and good references are provided), but the kind of careful broad review presented here is of significant value both to those working in dynamical systems as well as anyone interested in a fascinating mathematical story.

In very rough terms, KAM theory addresses subtle questions about the boundary between stability and chaos, and was originally motivated by issues from celestial mechanics, such as the ultimate stability of the solar system. The story begins with Poincaré and his efforts to understand perturbations of integrable Hamiltonian systems. It then extends over the better part of a century, beyond the lives of K, A and M.

KAM theory is unusual among mathematical theories in several respects. It is probably far less well known than other ideas of its stature. The names attached to it are a source of controversy: some say that Kolmogorov’s sketch of a proof had so many gaps it wasn’t a proof at all. Others argue the result is his entirely, that Arnold and Moser had only minor parts. Yet others say that Siegel’s name should be prominently attached to the result. Even the consequences of KAM theory are still argued: what does it really say about the stability of the solar system, and does it really invalidate the ergodic hypothesis and so throw statistical mechanics to the winds?

There are not many books like this one. It has so many diverse elements: mathematical history, biographical sketches, bits of mathematical gossip. A section including a fairly deep discussion of physical applications follows another on the KAM theory in context (Pros and cons: Underappreciated? Proofs are easy? Worth celebration? Oversold?)

While the book may appeal more to those with some background in the area, it has the feel of a long dinner conversation with mathematicians well-versed in the theory, reminiscing, gossiping and talking informally about the big ideas.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

  • 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