512 Prof. P. Ehrenfest on Adiabatie Invariants 



§ 9. Difficulties which arise by a passage through singular 

 motions. Aperiodic motions. 



One of these difficulties has already been mentioned at the 

 end of § 6. A difficulty of a slightly different form arises 

 when we pass in an adiabatie reversible way from the vibra- 

 tions in an anisotropic quasi-elastic field of force to those in 

 an isotropic field *. If we begin with an anisotropic field, 

 with potential energy 



^IWfi' + v/^t, .... (32) 

 it is usual to treat each cf the two principal modes of vibra- 

 tion according to Planck's method ; only those motions are 

 allowed for which the energies of the principal modes of 

 vibration satisfy the equations 



— = nji ; - = n 2 h (33) 



J/j v 2 



According to our hypothesis these equations must remain 

 unchanged if v 1 and v 2 converge infinitely slowly to the same 

 value. The field becomes isotropic, and the total energy 

 satisfies the equation 



= (n i + n 2 )h (34) 



e 



At the other hand, an isotropic field of force is a central 

 field, hence Sommerf eld's formulae can be applied here. 

 These give : 



Moment of momentum = mr 2 cf) = n-^- , . . (35a) 



Total energy — e= (n + n')hv. . . (356) 



The motions allowed according to both sets of conditions 

 .are not the same ; in the first place, we cannot see why the 

 moment of momentum (wdiich is not a constant in the aniso- 

 tropic case, but oscillates between the values ±2 \/n ± ?i 2 . It) 

 should converge to one of the distinct values given b}^ (35 a) 

 for the isotropic case. These oscillations become slower 

 and slower if vi and v 2 become more and more equal to each 

 other, hence which value is attained when we have arrived 

 at the isotropic case depends on a double limiting process. 

 A second discrepancy, to which Epstein has drawn my 

 attention, is the following : for a circular motion we must 

 have in equation ('65 b) n f = 0, in (34), however, ni = ?i 2 , 



hence in the latter case - can be equal only to even multiples 



* Comp. a remark made by H. A. Lorentz, Proc. Acad. Amsterdam 

 <(]912^ 



t The mass of the moving particle is supposed to be unity. 



