11 



(13) 



5 (A). The linear oscillator. Tlie parameter is here the frequency. 

 The solution is as follows: 



d fbV\ d /dF\ dq 1 



The mean value of the right-hand side is cja. Thus we obtain 

 the well-known adiabatic invariant Cyja. 



B. Body rotating about a fixed axis. Calling the moment of inertia 

 (the parameter) A and the moment of momentum p, we have: 



H=—p' = c, p = F = 1/2 Ac, V= CdqF=q[/2Air,. . (16) 



dV _c,q dV_Aq_ (^^\_'^i 



d fdV\ c, 



'' = -rXu) = -i <•«> 



which gives Ci^ = const., hence also p= T= 1-^2^4^!=: const. 



Applications. 



6. The cyclic system. We call cyclic those co-ordinates which do not 

 occur in the expression for the Hamiltonian function (ignorable co- 

 ordinates according to Thomson and Tait's terminology). They will 

 be indicated by qx{x = \,1, . . . . , k), the remaining, non-cyclic co- 

 ordinates by qi{X=zk -\- \, ^ + 2, . . . , , n). 



Hence we have 



H = H iqy, qx, px',a) p^, = — -— = px =z Cx . . (19) 



Oqx 



The characteristic function now will be 



V = :^ Cx q, -\- W (gy, c; cy; a) (20) 



.r 



We shall further assume that c„ is the energy-constant ; we then 

 obtain 



dV d\V dV dlV ■ 



— = qx + ^=tx — = ^=n . . . (20') 



OCx ocx Oc) ocy 



where all t, excepting /„, are constants and tn = t -\- const. 



