VI.78 George Birkhoff

b. Oversiel, Michigan, 1884; d. Cambridge, Massachusetts, 1944 Difference equations; differential equations; dynamical systems; ergodic theory; relativity theory

At the International Congress of Mathematicians in 1924 the Russian mathematician A. N. Krylov described Birkhoff as “the POINCARÉ [VI.61] of America.” It was an apt description and one that Birkhoff would have relished, for he was deeply influenced by Poincaré’s work, in particular his great treatise on celestial mechanics.

Birkhoff studied first at Chicago under E. H. Moore and Oskar Bolza, and then at Harvard under W. F. Osgood and Maxime Bôcher. Returning to Chicago, he was awarded his doctorate in 1907 for a thesis on asymptotic expansions, boundary-value problems, and Sturm-Liouville theory. In 1909, after two years at Wisconsin under E. B. Van Vleck, he went to Princeton, where he formed a close association with Oswald Veblen. In 1912 he moved to Harvard and remained there, in professorial positions, until his sudden death in 1944. Birkhoff was steadfast in his support for the development of American mathematics, supervising forty-five doctoral students, including Marston Morse and Marshall Stone, and holding many distinguished positions within the scientific community. He was generally recognized, both at home and abroad, as the leading American mathematician of his generation.

Birkhoff first came to prominence with a memoir on the theory of linear difference equations (1911), and he continued to publish on the topic intermittently throughout his career. Related to this work were several papers on the theory of linear differential equations and a paper on the generalized Riemann problem (1913), which concerns complex functions defined by differential equations. (Until recently it was believed that the latter paper included a solution to Hilbert’s twenty-first problem, the Hilbert-Riemann problem, but in 1989 Bolibruch proved this belief to be mistaken.)

Throughout his life Birkhoff’s deepest interest in analysis lay in DYNAMICAL SYSTEMS [IV.14] and it was here that he enjoyed his greatest success. His overarching aim was to obtain a reduction of the most general dynamical system to a normal form from which a complete qualitative characterization could be deduced. As with Poincaré, the study of periodic motions was central to his work, and he wrote extensively on the THREE-BODY PROBLEM [V.33] as well as on questions connected with stability. Of his memoir on dynamical systems with two degrees of freedom (1917), which won the Bôcher prize in 1923, he is said to have remarked that it was as good a piece of work as he was ever likely to do. Another celebrated achievement was his proof of Poincaré’s topological “last geometric theorem,” the publication of which brought him immediate international acclaim (1913). (The theorem states that any one-to-one area-preserving transformation of an annulus that moves the boundary circles in opposite directions must have at least two fixed points, and its importance lies in the fact that its proof implies the existence of periodic orbits in the restricted three-body problem.) He introduced several new concepts into dynamical theory, including “recurrent motion” (1912) and “metric transitivity” (1928), and promoted the use of symbolism in dynamics (1935), the latter helping to pave the way for the formalized development of symbolic dynamics (the branch of dynamical systems invented by HADAMARD [VI.65] (1898) that deals with spaces consisting of infinite sequences of symbols) by Marston Morse and Gustav Hedlund at the end of the 1930s. His book Dynamical Systems (1927) was the first book on the qualitative theory of systems defined by differential equations. Awash with topological ideas, it provides a connected account of much of his earlier research.

Closely related to Birkhoff’s dynamical research was his work on ERGODIC THEORY [V.9]. Stimulated by the theorems of Bernard Koopman and VON NEUMANN [V1.91], Birkhoff presented his own ergodic theorem in 1931, a fundamental result both for statistical mechanics and for MEASURE THEORY [III.55], the proof of which combined Poincaré’s topological approach with the use of Lebesgue measure theory. (Roughly speaking, Birkhoff’s ergodic theorem states that for any dynamical system given by differential equations that possesses an invariant volume integral, there is a definite “time probability” p that any moving point, except those of a set of measure zero, will be in an assigned region v. In other words, if t is a total elapsed time interval and t* is the portion of time during which the point is in v, then lim t* /t = p.)

In the creation of physical theories Birkhoff advocated mathematical symmetry and simplicity above physical intuition. His books on relativity theory (which were among the first on the subject in English), Relativity and Modern Physics (1923) and The Origin, Nature, and Influence of Relativity (1925), were characteristically original and widely read. At the time of his death he was engaged in developing a new theory of matter (taken to be a perfect fluid), electricity, and gravitation, which he had first proposed in 1943 and which, unlike Einstein’s theory, was based on flat spacetime.

Birkhoff published in several other fields, including the CALCULUS OF VARIATIONS [III.94] and map coloring, and he was the coauthor (with Ralph Beatley) of a textbook of elementary geometry (1929). His paper (with O. D. Kellogg) on fixed points in function space (1922) provided a stimulus for the later work of Leray and Schauder.

Birkhoff had a lifelong interest in the arts and was fascinated by the problem of analyzing the fundamentals of musical and artistic form. In later life he lectured extensively on the application of mathematics to aesthetics, and his book Aesthetic Measure (1933) enjoyed popular success.

Further Reading

Aubin, D. 2005. George David Birkhoff. Dynamical systems. In Landmark Writings in Western Mathematics 1640–1940, edited by I. Grattan-Guinness, pp. 871–81. Amsterdam: Elsevier.