482 Dynamical Theory of Currents [OH. xvi 



As before each integrand must vanish. We have therefore at every instant 



w, <ft 



If the coordinates 61, Z , ... n are a ^ capable of independent variation, 

 this leads at once to the system of equations 



_ (,_i f 2 f ...n) ........... (508), 



fcw 30. 



while if the variations in 1} 2 are connected by the constraints implied 

 in equations (501), (502), ... we obtain, as before, the system of equations 



= 1,2, ...)... (609). 



The quantities lt 2 , ... are called the "generalised forces" correspond- 

 ing to the coordinates 1} 2 , .... 



Lagranges Equations for Impulsive Forces. 



552. Let us now suppose that the system is acted on by a series of 

 impulsive forces, these lasting through the infinitesimal interval from t = 

 to t = T. If we multiply equation (508) by dt and integrate throughout this 

 interval, we obtain 



ari*= T r^T. 



- dt = 



The interval r is to be considered as infinitesimal, and ^^- is finite. 



ou s 



Thus the second term may be neglected and the equation becomes 



dT f T 

 change in - = I S s dt ..................... (510). 



dO s J o 



r 



We call I s dt the generalised impulse corresponding to the generalised 

 ./o 



force 8 , and then, from the analogy between equation (510) and the equation 



change in momentum = impulse, 

 7\ r r 



we call - the generalised momentum corresponding to the generalised 

 dd s 



coordinate 6 8 . 



APPLICATION TO ELECTROMAGNETIC PHENOMENA. 



553. We have already obtained expressions for the energy of an electro- 

 static system, a system of magnets, of currents, etc., and in every case this 

 energy can be expressed in terms of coordinates associated with " accessible " 

 parts of the mechanism. We can also find the work done in any small change 

 in the system, so that we can obtain the values of the quantities denoted in 



