352 GENERALIZED COORDINATES 



283. The potential energy corresponding to any principal coor- 

 dinate ^ is tfi^p or, if we suppose ty l given by equation (198), is 



Similarly, the kinetic energy corresponding to this principal 

 vibration is &4 1 }, or 



Averaged over a very long time, the average values of cos 2 (kj e x ) 

 and of sin 2 (kj ej are each ^, so that the average potential and 

 kinetic energies are respectively 



and these are equal since kl = -- - Thus in any vibration the aver- 

 age kinetic and potential energies are equal. 



Unstable ^Equilibrium 

 284. Suppose now that any one of the coefficients in equation 



(195) is negative, say a r Let us put - = %%, so that \ will be 



PI 

 real. Equation (197) now assumes the form 



' 



and this has as solution 



showing that -^ increases indefinitely with the time, and does not 

 oscillate about the value -^ = 0. Thus the motion is unstable, and 

 we now see that the motion can only be stable provided all the 

 coefficients cc v a zt , a n are positive. In other words, 



For stable equilibrium the potential energy in the configuration 

 of equilibrium must be an absolute minimum. 



This is the result which has already been stated without proof 

 in 153. 



