Mathematics and Statistics Department,
The Faculty of Mathematics, Computing and Technology,
The Open University, Walton Hall, Milton Keynes,
Buckinghamshire, MK7 6AA, UK.
From early in his career Poincaré had been interested in the fundamental problems of celestial mechanics (such as the stability of the solar system), and many of the papers he published in the 1880s relate to his interest in the subject and to the three-body problem in particular. The problem figures prominently in his acclaimed four-part memoir on curves defined by differential equations (1881-1886), and Poincaré was quite clear about its motivating role. These papers are full of new ideas, many of which form the basis for results in his most famous work on the three-body problem, the memoir of 1890. In this memoir Poincaré developed a theory of periodic solutions that opened up an entirely new way of thinking about dynamical problems. It is renowned for containing the first description of what today we would describe as mathematical chaos and for providing the basis for his acclaimed Les méthodes nouvelles de la mécanique céleste (1892-1899). Poincaré also produced many papers relating to the problem—seeing to the needs of astronomers as well as to those of mathematicians—up to and including his final publication Sur un théorème de géométrie (1912).
In this talk I shall discuss Poincaré’s work on the problem and look forward towards some of the work his research inspired.