George D. Birkhoff

Surface transformations and their dynamical applications

A state of motion in a dynamical system with two degrees of freedom depends on two space and two velocity coiirdinates, and thus may be represented by means of a point in space of four dimensions. When only those motions are considered which correspond to a given value of the energy constant, the points lie in a certain three-dimensional manifold. The motions are given as curves in this manifold. One such curve passes through each point. Imagine these curves to be cut by a surface lying in the manifold. As the time increases, a moving point of the manifold describes a half-curve and meets the surface in successive points, P, pr . . . . . In this manner a particular transformation of the surface into itself namely that which takes any point P into the unique corresponding point pr _ is set up. This fundamental reduction of the dynamical problem to a transformation problem was first effected by POINCAR~, and later, more generally, by myself? In order to take further advantage of it I consider such transformations at length in the following paper, which appears here by the kind invitation of Professors MITTAG-LEFFLER and N6RLU~D. The dynamical applications are made briefly in conclusion. These bear on the difficult questions of integrability, stability, and the classification and interrelation of the various types of motions.

Proof of Poincaré’s geometric theorem

In a paper recently published in the Rendiconti del Circolo Matemático di Palermo (vol. 33, 1912, pp. 375-407) and entitled Sur un théorème de Géométrie, Poincaré enunciated a theorem of great importance, in particular for the restricted problem of three bodies; but, having only succeeded in treating a variety of special cases after long-continued efforts, he gave out the theorem for the consideration of other mathematicians. For some years I have been considering questions of a character similar to that presented by the theorem and it now turns out that methods which I have been using are here applicable. In the present paper I give the brief proof which I have obtained, but do not take up other results to which I have been led.t 1. Statement of the Theorem. Poincarés theorem may be stated in a simple form as follows: Let us suppose that a continuous one-to-one transformation T takes the ring R, formed by concentric circles Ca and Cb of radii a and 6 respectively (a > 6 > 0), into itself in such a way as to advance the points of Ca in a positive sense, and the points of Cb in the negative sense, and at the same time to preserve areas. Then there are at least two invariant points. In the proof of this theorem we shall use modified polar coordinates y = r2, x = 0 where r is the distance of the point ( x, y ) from the center of the circles, and 6 is the angle which a line from the center to ( x, y) makes with a fixed line through the center. The transformation T may be written then

General theory of linear difference equations

The theory of lirlear diflerence equations with ratiollal coefficients was in a very backward state until POINCARG t in 1882 developed the notion of asymptotic representation, and its application to this brallch of mathematics. Further important progress ill this direction has been made only very recently, notably by GALBRUN,? who by means of the Laplace transformation has investigated the properties of a set of fundamental solutions in the entire plane of the complex variable t. The ainl of the presellt paper is first to study the nature of these solutions to which I have been led by mealls of direct ulethods, and secondly to show that there exists a purely Riemannian theory of linear differellce equations. In particular certain rational functions of e27r8-lX are shown to play a part like that of the monodromic group constants of an ordinary linear differential equation. To the best of my knowledge, the importance of the functional standpoint in the field of difference equations was emphasized first by VAN VLECK in an inspiring series of lectures given at the University of Wisconsin in the spring of 1909, in which he conjectured the existence of sets of solutions analytic on either the left or the right side of the complex plane. On account of the extreme simplicity of the matrix notation, I have found it convenient to deal with a linear difference system of n equations of the first order

Singular points of ordinary linear differential equations

in which the functions X;ff(Z) are anaZytic or have a pote at Z oo, will transform (1) into a system of equations (1) with coefficients aij(z) of the same form (2), although q is not necessarily equal to q. Any such system of equations (1) will be said to be equivatent to (1 ) at z oo. The first part of the paper deals with the determination of the simplest system of differential equations which is equivalent to (1) at Z oo. In §1 an important Lemma on AnaZytic Fqbnctions is stated and proved. The applica-

What is the Ergodic Theorem

(1942). What is the Ergodic Theorem? The American Mathematical Monthly: Vol. 49, No. 4, pp. 222-226.

Natural isoperimetric conditions in the calculus of variations

1 Dynamic Systems with Two Degrees of Freedom, Trans. Amer. Math. Soc., 18 (1917). 2 For references to Morses work of 1925 and later see his book The Calculus of Variations in the Large, Colloquium Pub. Amer. Math. Soc., 18 (1934). 3 The theorem of Birkhoff (Dynamical Systems, p. 135) that any surface homeomorphic with a hypersphere has at least one closed geodesic depends for its proof upon a simple case of this extension. 4 It is understood that every sub-k-cycle of a k-cycle in 91k is also in 9k, and that homologous k-cycle in 9k having the same critical k-sets are to be regarded as identical. A similar remark holds for k-chains in S2k.

On a simple type of irregular singular point

provided that a proper determination of the constant c1 be made. In the present paper I wish to consider the solutions of (1) in the vicinity of x = so, but under the restriction that p ( x ) and q ( x ) have the form ( 2)t The method of attack is essentially the same as that which I have employed earlier in the consideration of singular points.t In this special case the results may be given a more striking form. The last two theorems have no analoguesin my earlier paper and are susceptible of wide generalization.

A mathematical critique of some physical theories

My purpose today is to review some of the mathematical-physical theories of the past and of the present, indicating briefly the nature of certain concepts upon which these theories rest as well as the attendant logical difficulties, and even proposing modifications which have occurred to me. Of course the subject is too large for an address of this kind, but interest in mathematical-physical ideas is very widespread, and their importance for both mathematicians and physicists is profound. I feel more justified in choosing this subject because it has occupied so much of my attention recently. 2. Geometry

Infinite products of analytic matrices

In a large part of the theory of functions of a single complex variable the matrix of analytic functions rather than the single analytic function must be taken as the fundamental element. This is certainly the case for the functions defined by linear difference and differential equations. The goal of the present paper is to show that the classical results of Weierstrass and Mittag-Leffler, treating of the formation of infinite products of functions with assigned singularities, admits of a natural extension to infinite products of matrices. The matrices considered will be square matrices of n2 elements and of determinant not identically zero. The concept of equivalence, which I have developed elsewhere,t lies at the basis of this extension: Let

Divergent series and singular points of ordinary differential equations

in a neighborhood of the point xi= = x,* = 0, in which all the functions Xi are supposed to be analytic, Xi(O, , 0) being zero for i= 1, 2, n, has been the subject of much study. This is justified by the applications which can be made of this form to various theories in analysis and in dynamics. Dulact has simplified the problem in many cases by reducing the equations (1) to simple reduced forms of which the integration can be made without difficulty. The integration of these reduced equations then furnishes the solution of the system (1) either in terms of a parameter or in the form of a system of integrals. Let mi be the roots of the so-called characteristic equation which, when written in determinant form, is aXi