23 



motion which satisfies (24) will be stationary, because in general it will be possible 

 for any such motion to find a set of coordinates in which it satisfies also (22). 

 It will thus be seen that, for a periodic system of several degrees of freedom, 

 condition (24) forms a simple generalisation of condition (10). From relation (8j, 

 which holds for two neighbouring motions of any periodic system, it follows 

 further that the energy of the system will be completely determined by the value 

 of /, just as for systems of one degree of freedom. 



Consider now a periodic system in some stationary state satisfying (24), and 

 let us assume that an external field is slowly established at a continuous rate and 

 that the motion at any moment during this process allows of a separation of vari- 

 ables in a certain set of coordinates. If we would assume that the efîect of the 

 field on the motion of the system at any moment could be calculated directly by 

 means of ordinary mechanics, we would find that the values of the /'s with respect to 

 the latter coordinates would remain constant during the process, but this would 

 involve that the values of the n's in (22) would in general not be entire numbers, 

 but would depend entirely on the accidental motion, satisfying (24), originally pos- 

 sessed by the system. That mechanics, however, cannot generally be applied directly 

 to determine the motion of a periodic system under influence of an increasing ex- 

 ternal field, is just what we should expect according to the singular position of 

 degenerate systems as regards mechanical transformations. In fact, in the presence 

 of a small external field, the motion of a periodic system will undergo slow varia- 

 tions as regards the shape and position of the orbit, and if the perturbed motion is 

 conditionally periodic these variations will be of a periodic nature. Formally, we 

 may therefore compare a periodic system exposed to an external field with a simple 

 mechanical system of one degree of freedom in which the particle performs a slow 

 oscillating motion. Now the frequency of a slow variation of the orbit will be seen 

 to be proportional to the intensity of the external field, and it is therefore obviously 

 impossible to establish the external field at a rate so slow that the comparative 

 change of its intensity during a period of this variation is small. The process which 

 takes place during the increase of the field will thus be analogous to that which 

 takes place if an oscillating particle is subject to the effect of external forces which 

 change considerably during a period. Just as the latter process generally will give 

 rise to emission or absorption of radiation and cannot be described by means of 

 ordinary mechanics, we must expect that the motion of a periodic system of several 

 degrees of freedom under the establishment of the external field cannot be determined 

 by ordinary mechanics, but that the field will give rise to effects of the same kind 

 as those which occur during a transition between two stationary states accompanied 

 by emission or absorbtion of radiation. Consequently we shall expect that, during 

 the establishment of the field, the system will in general adjust itself 

 in some un mechanical way until a stationary state is reached in which the 

 frequency (or frequencies) of the above mentioned slow variation of the orbit has a 

 relation to the additional energy of the system due to the presence of the external 



