Theorem
Angular momentum is conserved for a body subject to no net external torque.
Proof
Let a body consist of particles
at position vectors
The motion of each particle is determined by Newton's second law
or
Taking the cross product with
and summing over
gives
Now
since
We can write
The right hand side of this equation is the rate of change of angular momentum. Consider
by Newton's third law so if
then
and angular momentum is conserved.