516 Mr. J. M. Burgers on Adiabatic 



If we bring the system from a given state of motion, in 

 which the a and the constants of integration of the equations 

 of motion have definite values, in a reversible-adiabatic way 

 to another state of motion, the values of the constants of 

 integration will change. Let c=f(q,p,a,t) be a first 

 integral of the equations of motion, then, by means of 

 condition (iii.), we can prove that the total change ol c 

 during the adiabatic process amounts to 



&=.fi^.^.«fc = £?.&• ... (4) 



jda at oa 



(For the sake of simplicity we have supposed that only one 

 of the a is varied.) The dash over ~- denotes an appro- 

 priately determined mean value. According to condition (ii.) 

 we may take the mean with respect to the time, while 

 according to (i.) we may take the mean with respect to the 

 time in the undisturbed motion. 



An adiabatic invariant will now be defined as a function 

 of the constants of integration and of the parameters, the 

 total variation of which during the adiabatic process is 

 zero f . 



Suppose that the equations of motion of the mechanical 

 system are integrated. Then' it is always possible to express 

 the p as functions of the q, the a, and of n (canonical) 

 constants of integration a- . . . . * n . The " conditionally 

 periodical " systems are characterized by the following- 

 property : each momentum pk can be expressed as a function 

 which contains only qu together with the a ami a : 



p k =</Fk(<ik, a' . . . u n , a)% .... (5) 

 In connexion with supposition A (§ 1 ) the functions Fa- must 



* Cf. the paper in the Proc. Acad. Amsterdam. 



f If an integral c—f is independent of the a, c will be an adiabatic 

 invariant, e.g. the moment of momentum in the motion under the 

 influence of central forces. 



| Geometrical interpretation of this formula : If we draw a q-p- 

 diagram for the coordinate g Jc , the point (q, p) describes a closed curve, 

 the form of which is independent of the values of the other q&. 



