Chapter 68 - George David Birkhoff, Dynamical systems (1927)

Abstract

The first book to expound the qualitative theory of systems defined by differential equations, Birkhoff's Dynamical Systems (DS) created a new branch of mathematics separate from its roots in celestial mechanics and making broad use of topology. In DS, Birkhoff summarized more than 15 years of his own research along three main axes: the general theory of dynamical systems; the special case with two degrees of freedom; and the three-body problem in celestial mechanics. In the first two chapters, Birkhoff's treatment was traditional: he gave proofs for existence, uniqueness, and continuity theorems, and then discussed Lagrange's equations, Hamiltonian mechanics, and changes of variables. In the third chapter, solutions were studied in their formal aspects, that is, as power series about which questions of convergence were systematically laid aside as irrelevant to the matter at hand. The next chapter followed Poincaré's idea of investigating the stability of formal solutions near equilibrium or periodic motion. Finally, the fifth chapter presented the methods by means of which the existence of periodic motions could be established Dynamical systems stands strangely isolated among the mathematical literature of its time as a fundamental intermediary between Poincaré's perceptive work and the modern theory. Birkhoff's main ambition was that the final aim of the theory of motion must be directed toward the qualitative determination of all possible types of motions and of the interrelation of these motions..