Poincare, Celestial Mechanics, Dynamical-systems Theory and ~chaos~~*

As demonstrated by the success of James Gleick’s recent book [19871,there is considerable interest in the scientific community and among the general public in “chaos” and the “new science” which is supposed to accompany it. However, as usual, it is not easy to separate hyperbole from fact. In an attempt to do this, I will offer a precise definition of chaos in the context of differential equations: mathematical models which, since Newton, have played a vital role in scientific discovery. I will show how the classical problems of celestial mechanics led Poincaré to ask fundamental questions on the qualitative behavior of differential equations, and to realize that chaotic orbits would provide obstructions to the conventional methods of solving them. In a major paper which appeared almost exactly one hundred years ago, Poincaré studied mechanical systems with two degrees of freedom and identified an important class of solutions, now called transverse homoclinic orbits, the existence of which implies the system has no analytic integrals of motion other than the total (Hamiltonian) energy. I will explain these terms and outline the history of subsequent developments of these ideas by Birkhoff, Cartwright, Littlewood, Levinson and Smale, and describe how the ideas of Melnikov have made possible an “analytical algorithm” for the detection of chaos and proof of nonintegrability in wide classes of perturbed Hamitonian systems. I will discuss the physical implications of the malhematical statements that these methods afford. In the process, I will point out that, while there is a precise vocabulary and grammar of chaos, developed largely by mathematicians and stemming from Poincaré’s work, it is not always easy to use it in speaking of the real world. * An earlier version ofthis paper was delivered at a special A.A.A.S. session on the mathematical foundations of chaos at the Annual Meeting in San Francisco, January 15, 1989. The present version was delivered as the first part of the Mark Kac Memorial Lectures at Los Alamos National Laboratory on April 25, 1989. Parts of sections 3 and 4 are adapted from a paper in Chaos and Fractals (AMS, Providence, RI). The author thanks the Sherman Fairchild Foundation and the California Institute of Technology for their support during the preparation of this paper. Single orders for this issue PHYSICS REPORTS (Review Section of Physics Letters) 193, No. 3 (1990) 137—163. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check. Single issue price Dfl. 21.00, postage included. 0 370-1573/90/$9.45 © 1990 — Elsevier Science Publishers B.V. (North-Holland) POINCARE, CELESTIAL MECHANICS, DYNAMICAL-SYSTEMS THEORY AND “CHAOS”