LAGKANGE'S EQUATIONS 345 



and when the interval from ^ to t 2 is made vanishingly small, this 



dT 



measures simply the change in produced by the impulse. 



B0 1 



C** dT dT 



In the second term / - dt, the integrand -^r- is finite, so that 



when the interval of tune is supposed to vanish, this term will 

 vanish with it. Thus the equation becomes 



dT C tz 



change in = | 6 X dt. (183) 



d6 l Jti 



277. If F is an ordinary force acting impulsively through the 



/>* 2 

 interval ^ to ,, we call I Fdt the impulse. By analogy we call 



tA 



f 



Jti 



the generalized impulse, corresponding to the generalized coordi- 

 nate #!. Thus we have equation (183) in the form 



change in - = generalized impulse. 

 00, 



From analogy with the relation, 

 change in momentum of a particle = impulse on particle, 



dT 



we call T- the generalized momentum corresponding to the coor- 



dinate O r Thus with these meanings attached to the terms 

 "impulse" and "momentum," the relation 



change of momentum = impulse 

 is true in generalized coordinates. 



When our coordinates are x, y, z, the coordinates in space of a moving 

 particle, the generalized momenta will of course become identical with the 

 ordinary components of momentum. We have 



so that = mx, etc. 



dx 



