165 



qk = qko + ^ cti cos Q; -\- H /?/ si7i Qi . . . . (3) 



with analogous expressions for the j)]^. 



§ 2. Adiabatic disturbances of the system. 



As before we sliall assume that during the infinitely slow change 

 of the parameters tlie Hamiltonian equations for the original 

 coordinates and momenta q and p remain valid (see, however, 

 below, remark 4, a). In order to investigate how the variables Q 

 and P behave during such a process, it is simplest to consider into 

 what- expression the differential form : 



:i:pdq — IJ{q,p,a)(/t (4) 



changes by the transformation from the q, p to the Q, P'). As 

 remarked above this transformation is a contnct4ransformation', 

 hence as long as the a are not varied we have : 



^pdq— ^PdQ^ dW (5) 



dW being the complete differential of a function of the Q and P, 

 which may also contain the a. During the variation the a are 

 explicitly given functions of the time; the formula (5) has then to 

 be replaced by 



da 



:Spdg = :^ PdQ ^ F. — dt-\- DW (6) 



dt 



where : 



dW dW dWda 



nW=2-^dQ + 2-—dP + ---dr) ... (7) 

 oQ oP da dt 



F is ei function of Q, P and a, which — if the P have been 

 properly chosen — contains the Q only in the form of trigono- 

 metric functions -. 



Icos i 

 . \(m,Q, + .. .rrinQn). ... (8) 

 sin ) 



The proof of this proposition is given in § 3. 



Hence the differential expression (4) changes into : 



:E PdQ—\H'^{Q,P,a)- F.a]dt + DW. ... (9) 

 H*{Q,P,a) is obtained from H {q, p, a) by replacing the q and p 

 by their expansions in trigonometric series. Now the characteristic 

 property of the angular variables is that //* does not contain the 0. : 



H*= H*(P,a)') (10) 



The equations of motion for the Q and P are the canonical 

 equations derived from a Hamiltonian function, which is equal to 

 the coefficient of dt in the differential expression (9). ^) 



1) Whittaker, 1. c. p. 297. 



2) in order to simplify the formulae it is assumed that only one parameter a 

 is varied. 



3) Whittaker, 1. c. p. 407. 



