The KAM Story

The following sections are included:

• Dedication
• Preface
• Acknowledgments
• Contents

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 hypothesis 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…

This chapter introduces the basic ideas and terminology needed to understand KAM theory. It's best to see this material first, since I'll use much of this terminology (often anachronistically) when discussing the history. The reader who prefers a wholly non-technical survey of KAM theory and its significance may skip directly to Chapter 5. Though the material here is mathematical, the presentation is conversational and informal. More precise definitions of many terms may be found in the glossary (Appendix F), and suggested resources for additional background and deeper study are given in the reader's guide (Appendix D).

When we arrive later to a description of early KAM theory, the principal players will be C.L. Siegel, A.N. Kolmogorov, V.I. Arnold, and J.K. Moser. From the longer historical view leading up to KAM however, the principal players are I. Newton and H. Poincaré, with a supporting cast of many others. In this first part we try to make sense of the context of KAM theory in the largest possible historical setting.

In this chapter, I outline the breakthroughs that resulted in KAM theory and resolved the slow crisis described above. It goes without saying that the mathematicians involved—Siegel, Kolmogorov, Arnold, Moser—were among the leaders of their respective generations, and that KAM theory was only one aspect of their work, though no doubt a memorable one. Kolmogorov in particular seems to have had extraordinary mathematical vision, transforming each of the several fields (probability, turbulence, HPT, information and complexity theory) in which he took an interest.

In this chapter, I give a mostly non-technical overview of how KAM theory fits into its broader setting: what it says in simple terms, why it's important in mathematics, physics, and the history of ideas, and what enthusiasts and detractors say about it.

The main focus of this book is classical KAM theory. But a perspective view of KAM can only be had by knowing where it sits within Hamiltonian perturbation theory (or HPT). For this reason, I give here a rough overview of HPT…

This chapter sketches out a few applications—and consequences—of KAM theory in physics. I devote the most space here to celestial and statistical mechanics, since those subjects are closely linked with the earlier narrative in Chapter 3. The other material is very cursory, and refers the reader to more detailed references for in-depth treatments…

The following sections are included:

• Appendix A: Kolmogorov's 1954 paper
• Appendix B: Overview of Low-dimensional Small Divisor Problems
  • The linearization problem
    • From Schröoder's functional equation to the Siegel center problem
    • Refinements and optimal conditions for the Siegel problem
  • Mappings of the circle
• Appendix C: East Meets West — Russians, Europeans, Americans
  • Cultural stereotypes in mathematics
  • Cultural and stylistic tensions
  • Cultural cross-currents in KAM theory
• Appendix D: Guide to Further Reading
  • General references on KAM
    • Original KAM articles, and priority
    • Accessible proofs of KAM theorems
    • Books on KAM theory (what books?)
    • Review articles, monographs, & book chapters on KAM
    • Expository, historical, & other sources on KAM
  • Mathematical background
    • Dynamical systems and ODEs
    • Classical mechanics and Hamiltonian dynamics
    • Ergodic theory
  • Chaos theory
    • The popular side of chaos
    • A chaos debate
    • The aftermath of popular chaos theory
  • History
    • The special nature of history of math & physics
    • Early history of mathematics and astronomy
    • Between Newton and Poincaré
    • Weierstrass and Poincaré's time
    • The Painlevé conjecture & the n body problem
    • The Soviet & Russian schools of dynamical systems
    • History of dynamical systems in general
  • Biography
    • General biographical sources
    • The principals
  • Applications of KAM (and Nekhoroshev) theory
    • Applications to celestial mechanics; stability
    • Applications to statistical mechanics, ergodic theory
    • Other applications
  • Mathematical topics related to classical KAM theory
    • Low-dimensional small divisor problems
    • Aubry-Mather & weak KAM theory, KAM for PDE
    • Nekhoroshev theory
    • Arnold diffusion
  • Culture, philosophy, Bourbaki, etc.
• Appendix E: Selected Quotations
• Appendix F: Glossary

The following sections are included:

• Bibliography
• Index