WOKK 



For instance, if the potential energy is of the form 



it will be found that all the differential coefficients vanish in the 

 configuration given by x = 0. On drawing a graph of the function 

 W it appears that- the equilibrium is stable. 



It may be that all the differential coefficients of W vanish 

 because W is a constant throughout the whole of a range sur- 

 rounding the configuration under consideration. If this is so, 

 the system may be displaced, and there will be no force tending 

 to move it from its new configuration every configuration is 

 one of equilibrium. Equilibrium of this kind is called neutral 

 equilibrium. 



A case of neutral equilibrium has already occurred in Ex. 2, p. 179, a 

 door free to swing about a vertical line of hinges. A second case is that 

 of a sphere rolling on a horizontal plane. 



Systems possessing Several Degrees of Freedom 



153. So far we have considered only systems which are limited 

 to moving through a single series of configurations systems 

 with only a single degree of freedom. The determination of the 

 stability or instability of a system having more than one degree 

 of freedom is a more complex problem. 



If the potential energy is absolutely a minimum in a position 

 of equilibrium, so that every possible motion involves an increase 

 of potential energy, then the equilibrium is stable. This obviously 

 can be proved by the same argument as has served when there is 

 only one degree of freedom. 



If the potential energy is not an absolute minimum, that 

 to say, if displacements are possible in which the potential ener^ 

 decreases while moving away from the position of equilibrium, 

 then the configuration is one of unstable equilibrium. This will 

 proved later. It cannot be proved by the methods used in this chaj 

 ter, and so we defer the question until later (Chapter XII). 



