508 Prof. P. Ehrenfest on Adiabatic Invariants 



we must demand that p can take no other values than 



P=0, ±£, ±2 . 1. . . . (17) 



Remark. — The discussion sketched in the preceding lines 

 wants to be developed more sharply, as the adiabatic trans- 

 formation passes through a singular unperiodical motion, 

 which forms the limit between the oscillatory and the rota- 

 tory motions. It is necessary to analyse more precisely the 

 connexion between the adiabatic invariants for both types of 

 motion. 



§ 7. Connexion with Sommerf eld? s formulas for systems of 

 more than one degree of freedom. 



"We will show that the quantum formulae, which Sommerf eld 

 has given for the motion of a point in a plane about a New- 

 tonian centre of attraction, satisfy the adiabatic hypothesis. 



Let % (r, a 1? a 2 . . .) be the potential of a central attractive 

 force. The differential equations/ff the plane motion of a 

 point, written in polar coordinates, have the form 



mr-mr^ + ^=0, (18 a) 



at 



j t (mr»0) = O (18 b) 



From (18 b) we see immediately that the moment of mo- 

 mentum is an invariant against a change of the parameters 

 a l} a 2 . . . 



mr 2 (f) = p 2 = adiabatic invariant . . (19) 



Eliminating <j> from eq. (18 a) with the aid of (19), we get 



«*=&-*/< (20) 



mr 6 dr v y 



This equation has the same structure as the differential 

 equation for the motion of a point, which oscillates along a 

 straight line under the influence of a potential 



2 



^ =+ W+%(n^« 2 ) . . . (21) 



between two limiting values of r (r A >r B >0). But, accord- 

 ing to §§ 4 and 5, this periodical motion of one degree of 

 freedom possesses the adiabatic invariant 



2T" (T 



— - = \ \dqi dpi = adiabatic invariant. (22) 



