THE CANONICAL EQUATIONS 357 



Thus Lagrange's equations may be written as 



du. cL 3(T W) d (T 1 + W) 



-er 



6) 



while, by equation (202), - * = 



If we write H = T' + W, these equations assume the symmet- 

 rical form 



d0, m 



^L- _l, 

 dt du, 



du, m 



-j7 = -W^ c ' I4W 



288. This is known as the canonical form of the dynamical 

 equations. The function H is called the Hainiltonian function, and 

 since H = T' + W, we notice that H is the total energy expressed 

 as a function of the coordinates X , 2 , , 6 n and of the momenta 

 u l9 u 2 , -., u n . 



The canonical form is the simplest and most perfect form in 

 which the generalized dynamical equations can be expressed. For 

 this reason the canonical system of equations forms the starting 

 point of a great many investigations in higher dynamics, mathe- 

 matical physics, and mathematical astronomy. 



289. We may appropriately terminate the present book by giv- 

 ing illustrations of the use of generalized coordinates from two 

 branches of mathematical physics. 



Illustration from hydrodynamics. Let a solid of any shape be in a stream of 

 water flowing with uniform velocity V. If the solid is at a sufficient depth from 

 the surface, its presence will not disturb the flow at the surface, and the only 

 disturbance in the flow of the water will be in the neighborhood of the solid. 

 It can be proved, from elementary hydrodynamical principles, that there is 

 only one way in which the water can flow past the solid. Hence it follows 

 that the kinetic energy of the flow of the water is given by 



