37] GENERALIZED VECTOR -COMPONENT. 117 



of definition 42), for the component of the velocity of the r ih particle 

 according to q s , 



4 9 ) *+*+ 



Now we have by 32), dividing by dt, 



--//* 



The derivatives -~ contain only the coordinates q, not the velocities q', 

 which we see enter linearly, accordingly 



Making use of this relation, the expression 49) becomes 



Thus we find that the component of the velocity of any particle 

 according to the coordinate q s is equal to one -half the rate of change 

 of the square of its velocity as we change the velocity q'^ 1 ). This 

 result is not of itself of great physical importance, but leads us to 

 one that is. Inasmuch as the momentum is the important dynamical 

 quantity, multiplying by the mass of the particle we find 



m^f! + **;fj + m,*;g - ~ 



or the component of the momentum of a particle according to any 

 coordinate is the rate of change of its kinetic energy as we change 

 the corresponding velocity. Summing for the whole system, 



. 52) 



that is, the component of the momentum of a system according to 

 any generalized coordinate q s is the rate of change of kinetic energy 



1) It is to be observed that this "component" is not what we have called 

 the velocity q' g . 



