150 



where during tlie integration ^^. moves up and down once between 

 its limits (jik being an arbitrary integral number). 



Now Prof. P. Ehrenfest ^) has pointed out the great interest for 

 the theory of quanta of the so-called Adiabatic Invariants, i.e. 

 quantities the value of which does not change if the system is trans- 

 formed in an adiabatic way (definition by Ehheneest. I.e. and below 

 § 1) from one state of motion to another. He has shown that for 

 rigorously periodic systems the integral of action, extended over a 



1 

 full period P = -: 



V 



P 



2T 



dt .2T= p.2r= — 







does not change its value during an adiabatic variation of the system; 

 and also that both the quanta-formulae intioduced by Sommerfeld 

 for the elliptic motion are related to adiabatic invariants. As 

 Prof. Ehrenfest has already remarked it would be very interesting 

 to inquire whether the above mentioned quantities Ik are also adia- 

 batic invariants. In the following lines I will try to show that this 

 is the case. 



§ 1. General considerations about the adiabatic alteration o f a system. 



Suppose that the mechanical system nnder consideration possesses 

 n degrees of freedom ; the coordinates will be denoted by q^ . . . qn ; 

 the momenta by p^ . . . p„. H be the Hamiltonian function, expressed 

 in terms of the q and p. For the present we will only suppose that 

 no coordinate or momentum can increase indefinitely, but that all 

 of them will remain between certain limiting values (to be deduced 

 from the equations of motion) {supposition A). ^) 



In the function H besides the q and p aeYidan paramaters a oqqmv : 

 e.g. masses, electric charges, the intensity of a field of force. We 

 may imagine that during a certain time these parameters are changed 

 infinitely slowly. A reversible adiabatic variation of the system will 



1) P. Ehrenfest, these Proceedings Vol. XIX (1), p. 576, 1917. 



2) hi the problems treated by Epstein and others an azimuthal angle cp occurs, 

 which can increase indefinitely. The configuration of the system, however, is periodic 

 with respect to this coordinate ; an increase of 9 by 2t here takes the place of 

 the up and down motion of the other coordinates. Apart from this tlie further 

 treatment remains substantially the same. (It is also possible to introduce q = sin 'P 

 as a new variable — cf Charlier, Die Mechanik des Himmels I p. 112 — in 

 order to return to the general case). 



