A Classical Approach of Newtonian Mechanics

9 ANGULAR MOMENTUM 9.4 Angular momentum of a multi-component system

X

X X

fi j Fi^

mi

line of centres

ri Fj^

rj

mj

fj i

Figure 87: A multi-component system with central internal forces.

The equation of motion of the ith particle can be written

j i

p ̇i = (^) j
X
=1,N fij^ +^ Fi.^ (9.26)^
Taking the vector product of this equation with the position vector ri, we obtain
j/=i
ri × p ̇i^ =
Now, we have already seen that
j
X
=1,N ri^ ×^ fij^ +^ ri^ ×^ Fi.^ (9.27)^
ri × p ̇i =^
d(ri × pi)

. (9.28)
dt
We also know that the total angular momentum, L, of the system (about the
origin) can be written in the form

L =

i=1,N

ri × pi. (9.29)

Hence, summing Eq. (9.27) over all particles, we obtain
i/=j
dL
= ri
dt
i,j=1,N

× fij (^) + ri
i=1,N
× Fi. (9.30)