and the Theory of Quanta. 505 



In the case of a simple harmonic motion of one degree of 

 freedom we know that the mean of the kinetic energy is 

 equal to the mean of the potential energy ; hence both of 

 them are equal to half the total energy. So here we have 



c 



- = adiabatic invariant (5) 



v 



§ 5. Geometrical interpretation of the adiabatic invariant — 

 in the phase-space (q-p space). 



To get a connexion with the formulae used by Planck, 

 Debye, Bohr, Sommerfeld and others to introduce the 

 quanta, we will avail ourselves of a transformation of 

 the integral of Action, to w^hich Sommerfeld has drawn 

 attention * : 



( dt. 2T=\ dt%2Jhqh=2\dqh.pk=t\\dp h dq h . 

 Jo Jo h h J h JJ 



(6) 



Hence 



2T 



I^a^a (7) 



The double integrations at the right-hand side have the 

 following meaning : When the system performs its periodic 

 motion, its phase-point describes a closed curvef in the 

 2n-dimensional q-p space, and its n projections on the 

 two-dimensional surfaces^ (#i,_pi), (q*, P2) • • • (q n , p n ) de- 

 scribe n closed curves, jj dp},dq h is the area of the region 

 enclosed by the h th projection curve. 



~2T 

 Remarks. — A. The numerical value of — is not changed 



v ° 



if we pass to another system of coordinates for the descrip- 

 tion of the motion. Hence also the numerical value of the 

 right-hand side of equation (7) is independent of the system 

 of coordinates used. 



B. Systems exist possessing the following property : 

 with a suitable choice of the system of coordinates not only 

 is the total sum at the right-hand side of (7) an adiabatic 

 invariant, but each separate integral Jj dp k dqh is an invariant. 

 Compare the example of § 7. 



* A. Sommerfeld, Sitzungsber. d. bayr. Ahad, 1916, pp. 42-5-500 (§ 7). 

 t This expression must be altered in some way, if any of the co- 

 ordinates he angles which increase by 2?r in each period. 



Phil. Mag. S. 6. Vol. 33. No. 198. June 1917. 2 N 



