220 MOTION OF A VARIABLE SYSTEM. [391. 



Similarly, it will be seen in every other case that a rigid body 

 has as many independent equations of motion as it has degrees 

 of freedom, or as it requires variables to fix its position. 

 These variables may be called the co-ordinates of the rigid body. 

 Thus a free rigid body has 6 co-ordinates corresponding to its 

 6 degrees of freedom and 6 equations of motion ; we might take 

 as such co-ordinates the co-ordinates x, j/, ~z of the centroid and 

 Euler's angles 0, (/>, -v/r. 



391. These considerations can be generalized so as to apply 

 to a general variable system of n points with k conditions. 

 Such a system is said to have $nk = m co-ordinates because 

 it has $nk = m independent equations of motion (Art. 385). 

 In other words, in the place of the 3 Cartesian co-ordinates 

 x y y, z of the n points, subject to k conditional equations, we 

 may introduce $n k=m independent variables, say q lt q^ 

 '" <7m> which are so selected as to satisfy the k conditions (i) 

 identically. These variables are called the Lagrangian, or 

 generalized, co-ordinates of the system. 



By the introduction of these new variables the equations of 

 motion (4) assume a form which is known as the second Lagran- 

 gian form. 



Suppose, for instance, that the system is subject to only one 

 condition, viz. that one point P l of the system should remain on 

 the surface of the ellipsoid 



If we select two new variables q^ q^ connected with x^ y l9 

 #! by the equations x l =acosq l , y l = b sin q l cos q^ z = 

 *: sin ^ sin 2 , the condition </> = o is satisfied identically in the 

 new co-ordinates g v 2 . Hence, by introducing q lt q% in 

 the place of x lt j/ 1? z lt the condition < = o is eliminated from 

 the problem. 



We now proceed to establish the equations of motion in the 

 second Lagrangian form, for a variable system of n points-, with 



