2 1 6.] LAGRANGE'S EQUATIONS. U^ 



The relations (4), (5) can again be shown to hold for each 

 of the three Lagrangian co-ordinates g v q^ q%. 



If the equations of motion 



mx=X, my=Y> m'z Z (15) 



be multiplied by 3/i/d^, d/ 2 /d<?i, dfjdq^ and added, we find 



where ' Q^X^+Y^Z^ 



dq l dg l dq l 



By the method of Art. 212, equation (16) reduces to the form 



^ar_ar_ 



Similarly we find 



Cv (j -L U J. S-* 



77/ ^ ^T == ^2 



The three equations (17) and (18) are the Lagrangian equations 

 of motion of a free particle. 



216. If there exists a force-function U for the forces X, F, 



Z, i.e. if 



Y dU v BU - dU 

 JL=- t y=, < = - 



dx dy dz 



we have 



_ + ^ + , = 

 1 dx d^ 1 dy dq l dz d^ l 



and similarly 



3U 



In this case, one of the three equations (17), (18) can be 

 replaced by the equation of kinetic energy 



where h is a constant. 



