100 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. 



from those of the actual motion, but if not they are traversed with 

 different velocities, so that at any rate the coordinates of the various 

 points are different functions of the time in the varied and in the 

 actual motion. 



In case the system is conservative, Hamilton's principle is 

 equivalent to another, less general, hut historically older. In this 

 form of statement of the principle, we compare two infinitely near 

 motions, but the second is not completely arbitrary, for instead of 

 associating together pairs of points x, y, z, x + Sx, y + dy, z -f #, 

 reached at the same instant in the two motions, and making dx, dy, dz 

 perfectly arbitrary, we assume that the variation takes place in 

 accordance with the equation of energy, 



5) T+W = h, 



so that we are to put 



8W=-dT. 



But if the equation of energy is to hold on the varied path as well 

 as on the unvaried, the kinetic energy of the system in any con- 

 figuration is determined, and thus the system may not be in that 

 configuration at any time we please, as that would involve arbitrary 

 velocities, and there is a restriction on the velocities due to the 

 determination of the kinetic energy for every configuration in the 

 motion. We will therefore give up the assumption that pairs of 

 points compared are reached at the same instant of time, in other 

 words we shall no longer assume that St = 0. No matter what the 

 independent variable may be, as functions of which we may express t 

 and all the coordinates, so as to compare the motions point by point, 

 we may use the principles explained in parenthesis on p. 80, which 

 will cause a certain modification of our result. If dt is not zero, 

 we can no longer put in the preceding demonstration, 



dSx _ ftdx 

 ~dT'' dt* 



but must write, as explained on p. 80, 



ddx _ ~dx ,dx, dSt 

 ~dT~ ~dt + ~di~dT' 

 dSy _ *dy dy^ dSt 

 dt dt "^ dt dt ' 



dSz _ *dz dz ddt 

 ~W' ~dt + ~dt~dT' 



We have thus to add to the right-hand member of equation 1), 

 the term 



2 T d st 



dt \ ' l 



