50, 51] CYCLIC SYSTEMS. 139 



be called the positional coordinates or parameters of the system. In 

 the example of 48 if we suppose the radial motion to be so slow 

 that we may neglect r n in comparison with r 2 cp' 2 we have 



184) T=ymrV 2 , 



and the system is cyclic, r being the positional, cp the cyclic co- 

 ordinate. In the case of a liquid circulating through an endless 

 rubber tube, the positional coordinates would specify the shape and 

 position of the tube. The positional coordinates will be distinguished 

 from the cyclic coordinates by not being marked with a bar. The 

 analytical conditions for a cyclic system will accordingly be, for all 



coordinates, either 



1Q .s 3T n 3T 



18o) ^ = or ^7=^ = 0, 



or if we use the Hamiltonian equations 78) 39 with the value of T 

 obtained by replacing the velocities by the momenta, which we shall 

 denote by T p , since the non- cyclic momenta vanish 



186) ^ = 0, and |5? = 0, 



for the cyclic coordinates, as before. We accordingly have for the 

 external impressed forces tending to increase the positional coordinates, 

 by 37, 60), 39, 80) respectively, the first term vanishing, 



TT) _d(l P +W) i) 



dT 



w- P- 



s ~ 



and for the cyclic coordinates 



A motion in which there are no forces tending to change the 

 cyclic coordinates is called an adiabatic motion, since in it no energy 

 enters or leaves the system through the cyclic coordinates. (It may 

 do so through the positional coordinates.) Accordingly in such a 

 motion the cyclic momenta remain constant. The case worked out 

 above was such a motion. 



In adiabatic motions the cyclic velocities do not generally remain 

 constant. In the above example, for instance, the cyclic velocity (p' 

 was given by 



A motion in which the cyclic velocities remain constant is called 

 isocydie. 



o rn O rji 



1) That - - = -o-^ may be seen by putting r = m in 144) , when the 

 parenthesis becomes T' 2T=T P . 



