132-133] GENERALIZED COORDINATES. 199 



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' q lt q^, ..., so that 



The kinetic energy can then be expressed as a homogeneous 

 quadratic function of the ' generalized velocity-components ' 

 q lt ,,..., thus 



2T=A ll q 1 * + A 22 q 2 2 +... + 2A lz q l q 2 + ............ (9), 



where, for example, 





A -s + + 



^1 12 2,m <-j -j- + -j --- ^ ---- h ~7~ j~ > 



[ogj rtg 2 dqi dq 2 dq l dq z ) 



The quantities A u , A^,..., A 13 ,... are called the 'inertia-coeffi- 

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

 the coordinates q lt q^, ....... 



Again, we have 



2X*Af+FAi 7 + ZA?)=Q 1 A0 1 + Q 8 Afc+ ...... (11), 



where, for example, 



^ i Y 



dq 1 dq^ ^ " dq l 



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

 force.' In the case of a conservative system we have 



If X', F', J^' 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 



and therefore 



