Research Threads Come Together: Harmonic Convergence

Abstract

Back in Chapter 2, I left off with the pre-KAM work by Waters and Ford (1966), and Zabusky and Kruskal (1965). Beginning in 1966, we find the news of KAM spreading across the physics community, influencing research directions, and forcing conclusions. In the first effort to bring the KAM theorem to bear on the FPU problem, Izrailev and Chirikov (1966) provided an explanation of the FPU boundedness (failure to approach equilibrium) in terms of a simplified, purely topological argument. Shortly thereafter, motivated in part by the work of Izrailev and Chirikov, Zabusky and Deem (1967) attempted to obtain equipartition of energy with the FPU Hamiltonian by exciting high-frequency modes initially, and then following the subsequent energy transfer using a continuum description. Then—in what is considered to be the historical marker for the general awareness of KAM in the physics community—Walker and Ford (1969) published a sizable article in Physical Review, demonstrating the properties of KAM using the Hénon and Heiles simulation as an example. In 1970, Ford again returns, this time with Gary Lunsford, to explore the irreversibility in weakly nonlinear systems based on arguments derived from Izrailev and Chirikov. Finally, in 1972 and 1973, Ford, Lunsford, Hénon, Zabusky, Stoddard, and Turner brought together the close relationship between the Hénon and Heiles Hamiltonian, the FPU problem, and the Toda lattice in a discussion of the ergodicity and integrability of these systems.

When the atoms are falling downward through empty space of their own weight, at indeterminate times and places they swerve a very little out of their downward course, just enough for you to call it a change of direction . If they did not swerve, then everything uiould fall straight down like raindrops through the void, no collisions would take place, and no impact of atoms upon each other, and Nature never uiould have produced anything at all.

Lucretius (Book II, lines 217–224

I first learned of KAM sometime before the Ford-Waters paper. Boris [Chirikov] got the details of KAM a good while before I did; some Moscow mathematician explained Arnold’s statement of KAM to him. Kruskal and Zabusky were initially hostile to Chirikov-Izrailev’s article, but then news of KAM got through to them. The Ford-Walker paper was an effort to popularize KAM and to make the Chirikov overlap criterion understandable via Héron-Heiles leuel curves.

Joseph Ford (Private communication)