﻿and its Relation to the Quantum Theory. 867 



Schwarzschild Angle Variables. 



The " angle variables " (Winkelkoordinaten) are intrinsic 

 coordinates which, are 2irt times the frequencies of vibration 

 of the system, and thus possess the characteristic properties 

 of being linear functions of the time such that alteration of 

 any one of them by an amount 2tt leaves the configuration 

 of the dynamical system unaltered. The two angle variables 

 for the family of orbits given in equations (17) and (18) are 

 therefore 2irv ] t and 2nrv 2 t, where v x and v 2 have the values 

 given in (23). The intrinsic momenta Pj and P 2 conjugate 

 to the angle variables Q x and Q 2 are constants defined by 

 the canonical equations 



dQi BH ^Q 2 _ BH 

 dt dlY dt : ~dP 2 ' 



where H is the Hamilton ian function (i. <?., the energy 

 regarded as a function of P x and P 2 ). The general solution 

 of the above equations can be shown to be * 







2n 







Equations (21) and (22) follow immediately on setting P : 

 and P 2 equal to integral multiples of — in accordance with 



LIT 



Schwarzschild's quantum conditions, which demand that 



,2tt 



PidQi = nih. 



k 



EhrenfesVs Adiabatic Hypothesis. 



Ehrenfest's adiabatic hypothesis states that motions 

 " allowed " by the quantum theory are transformed into new 

 " allowed " motions as the character of the dynamical system 

 is altered by changing very slowly some parameter 

 appearing in the energy. We shall take this parameter a 

 proportional to the perturbative force of repulsion between 



* For proof, use methods of Epsteiu (Ann. d. Phys. vol. li. p. 168). 



