1114 



The gknerai, method. 



1. Definition of adiabatic invariants'^). We consider a mechanical 

 system of n degrees of freedom, the equations of motion of which 

 must be written in the Hamiltonian form 



^H dH 



pi= - ^-, q, = ^-, 0=1,2, n) . . . . (3) 



Oqi dpi 



H is here a function of the /;, and qi . It must not contain t 

 explicitly. Moreover it is supposed to depend on certain external 

 co-ordinates, which we shall call the parameters Oj:. These parameters 

 may either retain constant values, in this case we have the iso- 

 parametric problem, or they may vary, which gives the rheo-para- 

 metric problem, or they may vary very slowly "), which is the 

 herpo-parametric or adiabatic pioblem, ro which we shall give 

 special attention. 



We shall make the following assumptions: 



1. None of the quantities }>{ or qi increases to infinity. The g-; are 

 confixed within tixed limits. 



II. During the time in which each qi goes to and fro many times 

 between its extreme values, the ax must change by an infinitely 

 small amount of the first order. Moreover each a^ must be approxi- 

 mately constant. Equations (3) must remain valid during the process. 

 It follows from these assumptions that the herpo-parametric problem, 

 will be obtained by putting ax = const, in the rheo-parametric problem 

 and then iaking for all the quantities the time-average in the 

 corresponding iso-parametric problem. 



In our discussion we shall confine ourselves to one parameter a. 

 This is not an essential limitation of the problem, but it simplifies 

 the formulae considerably. 



An adiabatic invariant is a function v of the integrationconstants 

 c,,c,, . . . . of the iso-parametric motion and of the parameter a, the 

 total "adiabatic" derivative of which with respect to a disappears: 



dv dv öu dc. bv dc, 



— = \ ^-| ^+..- (4) 



da da dcj da dc, dM 



where the horizontal line indicates the time-average. 



2. The iso-imrnmetric problem. In the equations of motion (3) we 



1) Comp. P. Ehrenfest 1. c. and J. M. Burgers 1. c. 



') Implicitly this condition will show itself in the fact, that Hamilton's function 

 only contains the parameters Ux itself and not the corresponding moments. 



