70, 71] CYCLIC SYSTEMS. 137 



Substituting this in the sum with respect to s in ^A we get 

 a ? A = 2,2, ^ 8?.' Sq r ' = 2.2 r Q r .8g;5g/. 



OQs 



But this expression represents ( 61 (3)) twice the energy of 

 B, possible motion in which the velocities would be $q s ', and must 

 therefore be positive for all values of &q $ ' , Sq r '. 



Accordingly S^ A > 0. 



The interpretation of this theorem for electrodynamics is 

 known as Lenz's Law*. 



71. Examples of Cyclic Systems. The expression for the 

 kinetic energy of the gyrostat worked out in 67 shows that 

 the system fulfils the conditions for a cyclic system if the velocity 

 6' is small enough to be neglected in comparison with the other 

 velocities. The forces acting have been already found, and we 

 can easily verify the theorems of the last two articles for this 

 case. 



A very simple case of a cyclic system is that of a mass m sliding 

 on a horizontal rod, revolving about a vertical 

 axis. Let us consider the mass concentrated at 

 a single point m at a distance r from the axis. 

 Let the angle made by the rod with a fixed hori- 

 zontal line be <j>, then the velocity perpendicular 

 to the rod is rty. The velocity along the rod Fm - 28 - 

 being r' ' , the kinetic energy of the body m is 



If we suppose the motion along the rod to be so slow that we 

 may neglect r' 2 



T= Jrnry 2 , 



and the system is cyclic, r is the positional, <f> the cyclic co- 

 ordinate. 



A system containing a single cyclic coordinate is called by 

 Helmholtz a monocyclic system. We have for the momenta 



0/77 O/7T 



* = 8? = 0. ^=^, 



* These Theorems are all given by Hertz, Principien der Mechanik, 568-583. 



