332 GENERALIZED COORDINATES 



of them are included in the sum. It follows that each term of the 

 sum must vanish. Thus we must have 



rS]- 



at every instant. 



271. At this point we have to consider two alternatives. It may 

 be that whatever values are assigned to B0 l} 80 2) - , B0 n> the new 

 configuration, specified by coordinates 



will be a possible configuration ; that is to say, will be one which 

 the system can assume without violating the constraints imposed 

 by the mechanism of the system. In this case the system is said 

 to have n degrees of freedom. 



If the system has n degrees of freedom, equation (165) is true 

 for all values of $0 l} S0 2 , , B0 n . For instance, it is true if we take 



/3 /3 _ Sk/D 5"/3 _ f\ 



Vi = e, V 2 = V 9 = = t>V n = U, 

 where e is any small quantity. In this case we must have 



cL d I uL \ I 



/ I I n 



oTT ~T, I T i I v 



, .. , d /dL\ dL . 



and therefore ( -7- -r = 0. 



\ 

 J 



dt \d0 



A similar equation will, of course, hold for each of the coordinates 

 #i> 2 > ' ' i @n' These equations are known as Lagrange's equa- 

 tions. There are n equations between the n unknown quantities 

 #i> 2 > ' ' > &n an( i their differential coefficients with respect to the 

 time. Thus they enable 1 us to find the way in which lt 2 , - , n 

 change with the time. To use the equations we require a knowl- 

 edge only of the function L, and therefore only of the kinetic and 

 potential energies of the system ; we do not need a knowledge of 

 the internal mechanism of the system. Thus the problem proposed 

 in 263 is solved, if we can solve Lagrange's equations. 



