General Dynamical Principles. 575 



The Conservation of Angular Momentum. 



We may next obtain an equation for the conservation of 

 the moment of momentum of an isolated system about any 

 axis. Let a be a unit vector along the axis, and consider 

 a particle with the mass m, moving with the velocity q. 

 The momentum of the particle wiil be ??iq, and i£ r is the 

 radius vector from any point on the axis to the particle, the 

 moment of momentum will be 



M=mq . (axr). 



Differentiating with respect to the time, we have for the 

 rate of change of the moment of momentum, 



r N.jjW+"i.(iXj) 



= {axr) .^(mq)+mq. (axq) 

 = (axr) .^(>q)-f 0. 

 Summing up for all the particles of the system we have 

 |2M=S(axr) . | („>q) = 2(axr) . (F + I). 



Similarly for the rate of change of the moment of 

 momentum of the continuously distributed mass contained 

 in the system we may write 



jUk dv=\(* x r) . j t (pq)tfi'=f(a x r) . (f+i)dv. 



; Adding these two equations and introducing the obvious 

 simplification which arises from the principle of the equality 

 of action and reaction, we have 



^XM+^rM^ = S(axr).F+f(axr).f^. . (4) 



But i(axr) .F + Uaxr) . f dv is the sum of the moments 

 of all the external forces acting on the system, and this 

 according to our equation is equal to the rate of change of 

 the moment of momentum of the system. If the system is 

 an isolated one, the moment of the external forces is zero, 

 and we have the principle of the conservation of the moment 

 of momentum. 



