4 HAMILTON'S EQUATIONS. 



energy is always to be treated in the differentiation as function 

 of the p's and q*s. 

 We have then 



* = ;fe* * l = -^ + Fl ' etc> (3) 



These equations will hold for any forces whatever. If the 

 'fetces^ &i*e dptterVative, in other words, if the expression (1) 

 j.s t an t exact differential, we may set 



where e q is a function of the coordinates which we shall call 

 the potential energy of the system. If we write e for the 

 total energy, we shall have 



e = P + e > (5) 



and equatipns (3) may be written 



*' = ;' * = -' etc - [I <> 



The potential energy (e 3 ) may depend on other variables 

 beside the coordinates q 1 . . . q n . We shall often suppose it to 

 depend in part on coordinates of external bodies, which we 

 shall denote by a x , # 2 , etc. We shall then have for the com- 

 plete value of the differential of the potential energy * 



de q = FI dq l . . F n dq n A 1 da^ A 2 da z etc., (7) 



where A^ A%, etc., represent forces (in the generalized sense) 

 exerted by the system on external bodies. For the total energy 

 (e) we shall have 



de=q l dp l . . . + q n dpn~Pidqi . . . 



p n dq n A l da-i A 2 da z etc. (8) 



It will be observed that the kinetic energy (e^,) in the 

 most general case is a quadratic function of the p's (or g-'s) 



* It will be observed, that although we call e the potential energy of the 

 system which we are considering, it is really so defined as to include that 

 energy which might be described as mutual to that system and external 

 bodies. 



