152 



dg dg 



dg z^ 2 ^ de -\- ^ da 

 de da 



(7) 



If . for such a function g : dg = O, it will be defined as an adiabatic 

 immriant ^). 



§ 2. If the equations of motion are completely integrated it is 

 always possible to express the momenta p^ - • • p,, as functions of 

 the q, the a and n constants of integration «' . . . .«" ^). In accordance 

 with what was said in the introduction we shall specialize to 

 systems where the expression of pk contains only the coordinate 

 qjc (together with the a and a): 



pk = \^Fjc{qk,a' .... «»,a) ')...... (8) 



(Supposition B). 

 In connection with supposition (A) of § 1 the functions Fk are 

 assumed to possess the following properties: 



(1) Each equation Fk{qk) = (^ has (at least) two simple roots 

 £jt and ija-; for values of qk between these roots Fk^O. 



(2) At a certain instant qk lies between Bk and i^k- 



It can then be shown that qk remains in this interval, and that 

 it performs a so-called libratkm ^) ^) fSupposition A'). 



The following integrals will now be introduced, which will be 

 called "phase-integrals" : 



k = I dqk . Pk = I dqk \/Fk (qk) = ^k («' • • • «" «) 



Jk = 



(9) 



^ Integrals c~ f which are independent of the a are themselves adiabatic 

 invariants (cf. form. 6). As an instance : in the motion under central forces the 

 integral of the moment of momentum. 



2) This may be accomplished 

 for instance by the integration of 

 the partial differential equation of 

 Hamilton-Jacobi. 



2) Geometrical interpretation of 

 this formula: If we draw a qp- 

 Ci diagram for the coordinate qk, the 

 ^ point (qk,J)k) describes a closed 

 curve, the form of which is in- 

 dependent of the values of the 

 other q. 



"*) Gf. Gharlier, Die Mechanik 

 des Himmels, (Leipzig 1902) I, 

 p. 86, 100. 



5) Gomp, note 2, page 150- 



