132-133] GENERALIZED COORDINATES. 



133. In the systems ordinarily considered in books on Dyna 

 mics, the position of every particle at any instant is completely 

 determined by the values of certain independent variables or 

 generalized coordinates ft, ft, , so that 



The kinetic energy can then be expressed as a homogeneous 

 quadratic function of the generalized velocity-components 

 ft, ft,..., thus 



2T=A n q 1 * + A 22 &+... + 2A l2 q l q 2 + ............ (9), 



where, for example, 



v 



A n = 2m T- + - + &amp;gt; 



Wft/ Uft/ ) 



^ + ##l 



-^ -, T^ 7 7 f &amp;gt; 



dq l dq 2 dq l dq z ) 



The quantities A llt A w ,..., A M ,... are called the inertia-coeffi 

 cients of the system ; they are, of course, in general functions of 

 the coordinates ft, ^2, ....... 



Again, we have 



2X*Af+FAi 7 + A?)=Q 1 A g r 1 + e 2 A0 a + ...... (11), 



where, for example, 



(12). 



% dft dq 



The quantities Q lt Q 2) ... are called the generalised components of 

 force. In the case of a conservative system we have 



If X , F , Z be the components of impulsive force by which the actual 

 motion of the particle m could be produced instantaneously from rest, we 

 have of course 



m|=Jr, w/7=r , mC = Z ........................ (i), 



and therefore 



