Adiabatic Invariants of Mechanical Systems. 515 



quantum-hypothesis, as used by Planck, Debye, Bohr, and 

 So mm erf eld, the quantity which is put equal to an integral 

 multiple of h is always an adiabatic invariant. As has 

 already been remarked by Ehrenfest, it would be of great 

 interest iif this could be proved to be the case with the 1^. 

 mentioned above. In this paper it will be shown that, if we 

 leave aside some special cases of degeneration, the I* are 

 really adiabatic invariants. 



1. General considerations on Adiabatic Disturbances of a 

 Mechanical System. 



We will consider a mechanical system of n degrees of 

 freedom ; the coordinates are denoted by q 1 . . . an ; the 

 momenta by p l . . . pn ; the Hamiltonian function may be 

 H(q,p, a). We will suppose that none of the coordinates 

 or of the momenta can increase indefinitely, and that the 

 coordinates move between fixed limits *. (Supposition A.) 



H does not depend only on q and p, but also on certain 

 parameters a, e. g. masses, electric charges, intensity of an 

 electric field, &c. Now during the motion of the system 

 these parameters may be changed by influences from without. 

 Such a variation of the parameters will be defined as a re- 

 versible adiabatic disturbance of the system, if it satisfies the 

 following conditions : — 



(i.) The variation is effected infinitely slowly with 

 respect to the motions of the system ; more 

 accurately : in a time so long that each co- 

 ordinate has gone up and down many times 

 between its limits, the a have changed by an 

 amount infinitesimal of the 1st order. 



(ii.) Approximately the -rr are constants. 



(iii.) During the process the equations 



dt ~dj> 7r 9 dt ~dq k 



remain valid f. 



(3) 



* In the problems studied by Epstein and others among- the co- 

 ordinates an aziruuthal angle <p occurs, which can increase indefinitely. 

 The configuration of the system is, however, periodic with respect to 

 this variable. An increase of <p by 2-x takes the place of the libration of 

 the other coordinates. With some slight modifications the con- 

 siderations given above are valid also for coordinates of this kind. 



t This is for instance always the case, if only those a which occur in 

 the function of forces are varied. 



