

Criterion of Potential Energy. 695 



7. It need hardly be pointed out that the potential energy 

 (energy of rotation) with which we are concerned in §§ 3-5 

 may lose its potential character if the conditions of the 

 motion are modified : for example, if the governor is acted 

 on by forces having a moment about its axis of rotation, so 

 that the angular momentum about that axis no longer remains 

 constant ; or if the axis A A is allowed to change its direction 

 in space. In such cases the kinetic character of the rotational 

 energy must be explicitly recognized in the dynamical 

 equations. 



8. In any conservative dynamical system let yfr, <j>, 0, . . . be 

 the coordinates which we employ to define the configuration 



at each instant, and let-v/r, <£, 0, . . . stand for the time-fluxes 

 of these icorking coordinates (as they may be called). Then 

 the kinetic energy of the system as ordinarily understood 

 may be expressed as a h.q.f.* of >/r, <£, 0, . . ., with coefficients 

 in general functions of i/r, $, 0, . . ., while the potential energy 

 is a function of ^r } cj>, 0, . . . only. As soon as we admit the 

 kinetic nature of the energy which we treat as potential, we 

 realize that in addition to the working coordinates yjr, (f>, 0, . . . 

 there must be others (say) %, ^', ^", . . . whose time-fluxes 



X> x' 9 X"> ■ * * are mv °l ve d in this so-called potential energy, 

 and which may be distinguished as "ignored coordinates "f. 



9. The Lagrangian function for any conservative dyna- 

 mical system is the difference of the kinetic and potential 

 energies, the energy (•'C) being expressed in terms of the 

 generalized velocities ^, <£, 0, . . . ; on the other hand, the 

 Hamiltonian reciprocal function is the sum of the kinetic and 

 potential energies, the kinetic energy being expressed in terms 

 of the momenta dE/d-f , dC/d^, ... In the first place, then, 

 it is apparent that potential energy is energy expressed in 

 the Hamiltonian form, in terms of momenta, not in the 

 Lagrangian form, in terms of velocities. 



10. Suppose at the outset that the entire energy T is 

 recognized as kinetic, and is expressed as a h.q.f. of \fr, <£, fi, . . . 

 and the %'s. T is then the Lagrangian function for the 

 system, and the equations of motion corresponding to 

 yjr, </>, 0, . . . are oi: the type 



d_ dT __ £T ^^ „. 



dt'^ ~df 



1 r , <J>, ®, . . . being the impressed forces of types corre- 

 sponding to yjr, (/>, 0, . . . respectively. 



* Here and below h.q.f. stands for " homogeneous quadratic function. " 

 t Thomson (Kelvin) and Tait's ' Natural Philosophy, Part 1. § 319. 



