504 Prof. P. Ehrenfest on Adiabatic Invariants 



retain their value during the transformation of a motion 

 /3(a) into a motion /3(V) related adiabatically to the former. 

 Indeed, from the hypothesis follows immediately the 

 property : 



If an adiabatic invariant Q, for the " allowed " motions 

 B{<2 }, belonging to the special values a 10 , a 20 . . . , possesses 

 the distinct numerical values 12', O" . . . , it possesses exactly 

 the same values for the " allowed " motions belonging to 

 arbitrary values of the parameters a 1? a 2 . . . 



§ 4. The adiabatic invariant — for periodic motions and ~ 



especially for harmonic motions 



Suppose that the system under consideration possesses the 

 following properties : — For arbitrarily fixed values of the 

 parameters a l9 a 2 . . . , all the motions that have to be con- 

 sidered are periodic, independently of the phases (^ l0 . . . q n0 , 

 qio • • • </no) the motion starts with. The period P may 

 depend in some way or other on the a l5 a 2 . . . and on the 

 beginning phase. 



Then the integral with respect to time of twice the kinetic 

 energy, extended over one period, is an adiabatic invariant : 



i 



dt . 2T = (3) 



In this formula B' denotes the difference in value for two 

 infinitely near, adiabatically related motions of the system. 

 (For the demonstration of form. 3 the reader is referred to 

 the original paper, Proc. Acad. Amsterdam,xxv. (1916) p. 412. 

 Putting the reciprocal of the period P equal to the fre- 

 quency v, and denoting the mean of T with respect to the 

 time by T, form. 3 expresses : 



2T 



— = adiabatic invariant. ... (4) 



* Comp. paper C, § 1, 2. Other instances of adiabatic invariants: If 

 the system possesses cyclic coordinates, the cyclic momenta are in- 

 variants. If the rotation of a ring of electrons is affected by an 

 increasing magnetic field, the sum of the moment of momentum and of 

 the electrohinetic moment is an invariant (Zeeman effect, magnetization). 

 If an increasing electric field acts on a hydrogen atom of Bohr, then the 

 component of the moment of momentum parallel to the lines of force is 

 an invariant. For changes of a central field of force it is the moment 

 of momentum. 



