40,41] HAMILTON'S PRINCIPLE GENERAL. 131 



Hamilton's Principle holds whether the system is conservative or 

 not, it is more general than the principle of Conservation of Energy, 

 which it includes. The principle of energy is not sufficient to 

 deduce the equations of motion. If we know the Lagrangian func- 

 tion we can at once form the equations of motion by Hamilton's 

 Principle, and without forming them we may find the energy. For 

 we have 



L = T - W, 



E=T+W, 



Accordingly 



87) E-2T-L 



so that the energy is given in terms of L and its partial derivatives. 

 If on the other hand the energy E is given as a function of the co- 

 ordinates and velocities, the Lagrangian function must be found by 

 integrating the partial differential equation 87), the integration 

 involving an arbitrary function. In fact if F be a homogeneous 

 linear function of the velocities, the equation 87) will be satisfied 

 not only by L but also by L + F. For, F being homogeneous, of 

 degree one, 



, cF 



Consequently a knowledge of the energy is not sufficient to find 

 the motion, while a knowledge of the Lagrangian function is. The 

 attempt has been made by certain writers to found the whole of 

 physics upon the principle of energy. The fact that the principle 

 of energy is but one integral of the differential equations, and is 

 not sufficient to deduce them, should be sufficient to show the 

 futility of this attempt. It is the infinite order of variability of the 

 motion involved in the variations occurring in Hamilton's Principle 

 that makes it embrace what the Principle of Energy does not. 



41. Principle of Varying Action. We shall now deal with 

 a principle, likewise due to Hamilton, somewhat broader than that 

 which we have hitherto called Hamilton's Principle or Principle of 

 Least Action, and furnishing a means of integrating the equations 

 of motion. In the principle of least action a certain integral, belong- 

 ing to a motion naturally described by a system under the action 

 of certain forces according to the differential equations of motion, 

 has been compared with the value of the same integral for a slightly 

 different motion between the same terminal configurations, but not 

 a natural motion and therefore violating the equations of motion. 

 Under these circumstances the principle states that the integral is 



9* 



