LAGKANGE'S EQUATIONS 329 



LAGRANGE'S EQUATIONS 



269. If the coordinates x lf y ly z lt etc., of every particle of 

 the system are known, we know not only the configuration of 

 the system but also the mechanism by which the different parts 

 of the system are connected. It may, however, be that we can 

 determine the configuration of the system, by knowing a smaller 

 number of quantities which do not give us a knowledge of the 

 mechanism. 



For instance, in our former illustration we imagined two ropes to hang 

 from an unknown machine, the ropes being connected in such a way that 

 a motion of one inch in the one invariably produced a motion of two inches 

 in the other. In this case the configuration is fully determined when we 

 know the single coordinate which measures the position of the end of the 

 first rope, but a knowledge of this coordinate does not imply a knowledge 

 of the mechanism connecting the ropes. 



Again, the position of a rigid body is, as we have seen ( 65), determined 

 by the values of sufficient quantities (six) to fix the positions in space of 

 three non-collinear particles of the body, but a knowledge of these quanti- 

 ties does not give us information as to the arrangement of the particles of 

 which the body is formed. 



Let 6 V 2 , , 6 n be a set of quantities such that when their 

 value is known, the configuration of a system of bodies is fully 

 determined. Then the quantities V 2 , , 6 n are called general- 

 ized coordinates of the system. 



270. Let Xy y, z be the, coordinates of any particle of the system. 

 Then x is fully determined by the values of V 2 , , n , so that 

 it is a function of these quantities, say 



If the system is in motion, all the quantities which enter in 

 equation (158) are functions of the time. We have, on differ- 

 entiation with respect to the time, 



dx^df^dO^ df^d^ df_ dj^ 



dt ~ d6 l dt + W 2 dt " dd n dt ' 



