54 



SCIENCE 



[N. S. Vol. LI. No. 1307 



means of appropriate changes of variables. 

 In consequence the coordinates of dynamical 

 systems admit of simple analytic representa- 

 tion for all values of the time. In particular 

 Sundman has proved that the coordinates and 

 the time in the problem of three bodies can 

 be expressed in terms of permanently con- 

 vergent power series, and thus he has " solved " 

 the problem of three bodies in the highly arti- 

 ficial sense proposed by Painleve in 1897. 

 Unf ortimately these series are valueless either 

 as a means of obtaining qualitative informa- 

 tion or as a basis for numerical computation, 

 and thus are not of particular importance. 



Prom early times the mind of man has 

 persistently endeavored to characterize the 

 properties of the motions of the stars by 

 means of periodicities. It seems doubtful 

 whether any other mode of satisfactory de- 

 scription is possible. The intuitive basis for 

 this is easily stated: any motion of a dy- 

 namical system must tend with lapse of time 

 towards a characteristic cyclic mode of be- 

 havior. 



Thus, in characterizing the motions of a 

 dynamical system, those of periodic tj^pe are 

 of central importance and simplicity. Much 

 recent work has dealt with the existence of 

 periodic motions, mainly for dynamical sys- 

 tems with two degrees of freedom. 



An early method of attack was that of 

 analytical continuation, due to Hill and Poin- 

 care. A periodic motion maintains its ident- 

 ity under continuous variation pf a parameter 

 in the dynamical problem, and may be fol- 

 lowed through the resultant changes. G. 

 Darwin, F. E. Moulton and others have ap- 

 plied this method to the restricted problem 

 of three bodies. Symmetrical motions can be 

 treated frequently by particularly simple 

 methods. Hill made use of this fact in his 

 work. 



Another method is 'based on the geodesic in- 

 terpretation of dynamical problems. This has 

 been developed by Hadamard, Poincare, Whit- 

 taker, myself, and others. The closed geodesies 

 correspond to the periodic motions, and the 

 fact that certain closed geodesies of minimum 

 length must exist forms the basis of the argu- 

 ment in many cases. As an example of an- 



other type, take any surface with the con- 

 nectivity of a sphere and imagine to lie in 

 it a string of the minimum length which can 

 be slipped over the surface. Clearly in being 

 slipped over the surface there will be an 

 intermediate position in which the string will 

 be taut and will coincide with a closed 

 geodesic. 



Finally there is a less immediate method of 

 attack which Poincare introduced in 1912, 

 and which I have tried to extend. By it the 

 existence of periodic motions is made to de- 

 pend on the existence of invariant points of 

 certain continua under one-to-one continuous 

 transformation. The successful application 

 of this method involves a preliminary knowl- 

 edge of certain of the simpler periodic 

 motions. 



Periodic motions fall into two classes which 

 we may call hyperbolic and elliptic. In the 

 hyperbolic case analytic families of nearby 

 motions asymptotic to the given periodic 

 motion in either sense exist, while all other 

 nearby motions approach and then recede 

 from it with the passing of time. In the 

 elliptic case the motion is formally stable, 

 but the phenomenon of asymptotic families 

 not of analytic type arises unless the motion 

 is stable in the sense of Levi-Civita. 



In a very deep sense the periodic motions 

 bear the same kind of relation to the totality 

 of motions that repeating doubly infinite 

 sequences of integers 1 to 9 such as 



. . . 2323 . . . 

 do to the totality of such sequences. 



In trying to deal with the totality of 

 possible types motion it seems desirable to 

 generalize the concept of periodic motion to 

 recurrent motion as follows: any motion is 

 recurrent if, during any interval of time in 

 the past or future of sufficiently long dur- 

 ation T, it comes arbitrarily near to all of its 

 states of motion. "With this definition I have 

 proved that every motion is either recurrent 

 or approaches with uniform frequency arbi- 

 trarily near a set of recurrent motions. 



The recurrent motions correspond to those 

 double sequences specified above in w'hich every 

 finite sequence which is present at all occurs 



