39] HAMILTON'S EQUATIONS. 127 



Let us now introduce into T the variables p in place of the 



variables q', so that T is expressed as a function of all the </'s 



and j?'s. Since 



38) ** 



inserting the values of q', in terms of the p's gives 

 72) T 



that is T is now expressed as a quadratic form in the ^s. We will 

 distinguish T when expressed in terms of the ^'s by the suffix^, T p . 

 We now have by Euler's Theorem, 



dT 



Since T p is identically equal to T, comparing with equations 38), 

 above we have by symmetry, 



thus the q"8 are linear forms in the p's given by 71). The two 

 identically equal functions, T, T p , having the properties 



dT dT 



75) Pr, 



are said to be reciprocal functions. 1 ) 



The expressions for the forces and potential energy are left 

 unaltered. Let us now make use of Hamilton's principle with this 

 choice of variables. Before performing the variation it will be 

 advantageous to introduce in the integral to be varied instead of the 

 Lagrangian function, L = T W } the Hamiltonian function, H= T+W, 

 by means of the relation 



76) L = 2T-H. 



T and H are both to be expressed as functions of the variables q 

 and p, both of which depend upon the time t in a manner to be 

 found by integrating the differential equations of motion. 

 Hamilton's principle then takes the form 



77) dl(2T -H)dt = d I Z r (p r ql - H)dt 



J J 



t to 



tl 



C^ri / f dH dH \ 



= I ^ r I q r dp r -(- p r dcir ^ $p r ^ o q r I (*^. 



/ ^mJ > ^JP r ^ ^r ' 



#0 



1) Webster, Theory of Electricity and Magnetism, 63, 64. 



