Criterion of Potential Energy, 



693 



2. The potential energy of a conservative dynamical 

 system is sometimes defined as that portion of the total 

 energy which is a function of the coordinates of the system 

 only, and is independent of the rates of change of those 

 coordinates. Such a definition, however, is too inclusive, for 

 in the energy-expressions of many dynamical systems there 

 are terms which, although independent of the time-fluxes of 

 the (working) coordinates *, do not fall into the category of 

 potential energy. 



3. Before treating the question more generally, it will be 

 convenient to consider an example. Fig. 1 represents what 



Fiff. 1. 



may be shortly called a governor, though we are not here 

 concerned with its capacity for " governing." The spindle 

 A A is frictionlessly journalled in a fixed frame B B, and the 

 only other freedom of the system corresponds to a motion of 

 the block C lengthwise of the spindle, with accompanying 

 motion at the pivots E, "F, G. For convenience of de- 

 scription, suppose the axis A A to be vertical, and as coor- 

 dinates of the system take x defining the azimuth of the 

 governor, and z a length measured vertically downwards 

 from a fixed horizontal plane to some point of the block G. 

 Using dots over the coordinates to signify their time-fluxes, 

 the kinetic energy may be expressed in the form 



E=nis-+ux 2 - (i) 



where I, the moment of inertia about A A, is a function of z 7 

 and M (a coefficient of the same kind as a mass) is also a 

 function of z. 



4. Let the only external force acting be a vertical force Z 



* Of. § 8 below. 



