George Birkhoff’s forgotten manuscript and his programme for dynamics

Abstract

In 1912 George Birkhoff created a sensation with his proof of Poincaré’s so-called “last geometric theorem.” He followed it with prize-winning papers on “The restricted problem of three bodies” (1915) and “Dynamical systems with two degrees of freedom” (1917). Many of the essential ideas from these papers can be found in his book Dynamical Systems (1927). At the end of the 1920s, Birkhoff began to draw up a programme of research on unsolved problems in dynamics, and in 1941 presented his ideas at the 50th anniversary celebration of the University of Chicago. Soon afterwards a summary of his lecture was published. At the time of his death in 1944, he left unfinished a manuscript of a revised and extended version of his lecture. In this paper I describe Birkhoff’s work leading up to this manuscript before describing the contents of the manuscript itself.