The Kolmogorov-Arnold-Moser Theorem: “Here Comes the Surprise”

Abstract

Around 1610, Galileo began the formalization of terrestrial mechanics, while Kepler did the same for celestial mechanics Sir Isaac Newton saw fit to combine these into one subject, unifying them with his calculus. Newton himself worked out the equations of motion for the two-body problem in order to demonstrate his method (1687). In 1788, the great Italian–bench mathematician Joseph Louis Lagrange, who had been working with Euler on the calculus of variations, was able to formalize his brilliant generalized method of finding the equations of motion for any mechanical system—the same method we use today. It was also Lagrange, and then Laplace, who recognized the difficulty of understanding the motion of three bodies, and so arose the question of the stability of the Solar System. In 1834, in Dublin, Sir William Rowan Hamilton placed position and momentum on equal footing as the canonical variables of dynamics, and he also showed that the variational principle beneath dynamics was the same as the principle of least time in optics, thereby uniting the two disparate formalisms at the theoretical level. Soon after Hamilton’s contribution, Joseph Liouville proved a theorem implying that if energy was conserved in a system, then any volume of initial conditions in phase space must be conserved throughout the evolution. Liouville’s theorem holds even if the system is ergodic and highly complex.

In order to connect familiar with unfamiliar territory, in other words to make the unfamiliar familiar, to be able to find his way back, the explorer cuis a blaze precisely on the boundary of the known world. ... The region from which the blaze is visible is unfamiliar in the sense that the explorer has never seen it before, bui familiar in the sense that he still knows his way back from it, and can safely venture there.

Frederick Turner (1978, p. 625)

But here comes the surprise ... All the isolated points correspond to one and the same trajectory, just as the points on one of the closed curues; but they behave in a completely different way. ... They seem to be distributed at random, in an area left free between the closed curves. Most striking is the fact that this change of behavior seems to occur abruptly across some dividing line in the plane.

Michel Hénon and Carl Heiles (1964, p. 76)