43 



n. Denoting the value of Ihe energy of the undislurhed system in sucii ;i slati' l)y 

 En, and the value of the energy of the perturbed system in a state belonging to 

 the multitude under consideration by Zi„ -f (i, we gel from (42) 



d& 



S 1 



= y\'oi< à h (43) 



for the energy difference between two neighbouring states of this multitude. Since 

 this relation has the same form as (29), we see consequently that by putting /,, ... h-i 

 equal to entire multipla of h, as claimed by the conditions (22), we obtain exactly 

 the same relation between the additional energy (ï and the small frequencies o^ , 

 impressed on the system by the external field, as that which holds between the 

 total energy and the fundamental frequencies in the stationary states of a condi- 

 tionally periodic system of s — 1 degrees of freedom. 



As a simple illustration of tliese calculations let us consider tlie system consisting 

 of a particle moving in a plane and sutijecl to an attraction from a fixed point, wliicli varies 

 proportional to tlie distance apart. If undisturbed, tlae motion of tliis system will be jjeriodic 

 independent of tlie initial conditions, and the particle will describe an elliptical orbit with its 

 centre at the fixed point. Moreover the equations of motion of the undisturbed system maj' 

 be solved by means of separation of variables in polar coordinates, as well as in any set of 

 rectangular coordinates. In the first case we have, taking for q^ the length of the radius vec- 

 tor from the fixed point to the particle and for r/, the angular distance of this radius vector 

 from a fixed direction, x^ ^ 2 and x^ = 1, while in the second case we have z^ = z, = 1. In 

 the presence of an external field the orbit will in general not remain periodic, but will in the 

 course of time cover a continuous extension of tlie plane. If the external field is sufficientlv 

 small, however, the orbit will at any moment only difTer little from a closed elliptical orbit, 

 but in the course of time the lengths and directions of the principal axes of this ellipse will 

 undergo slow variations. In general the perturbed system will not allow of separation of 

 variables, but two cases obviously present themselves in which such a separation is still 

 possible; in the first case the external field is central with the fixed point as centre, and a 

 separation is possible in polar coordinates; in the second case the external field of force is 

 perpendicular to a given line and varies as some function of the distance from this line, and 

 separation is possible in a set of rectangular coordinates with the axes parallel and per- 

 pendicular to the given line. In the first case the perturbations will not affect the lengths of 

 the principal axes of the elliptical orbit and will only produce a slow uniform rotation of 

 the directions of these axes, while in the second case the lengths of the principal axes as well 

 as their directions will perform slow oscillations. It will consequentlv be seen that, by fixing 

 tlie stationary slates of the perturbed system by means of the conditions (22), the cycles of 

 shapes and jiositions which the orbit of the particle will pass through in the stationary slates 

 will be entirely different in the two cases. In both cases, however, it will be seen that the 

 frequency o = oij — z^ (Ug will be equal to the frequency with which the orbit at regular 

 intervals re-assumes its shape and position. By fixing the stationary states by (22) we obtain 

 therefore, as seen from (43), in both cases that the relation between this frequency and the 

 additional energy of the system due to the presence of the field will be the same as the 

 relation between energy and frequency in the stationary stales of a system of one degree of 

 freedom; and it will be seen that the above considerations offer a dynamical interpretation 

 of the characteristic discontinuity involved in the application of the method of separation of 

 variables to the fixation of the stationary states of perturbed periodic .systems'). 



') 111 tills connection it may be of inteiest to note tliat the possibilitj' of a rational interpretation of 



