## Proof of Conservation of Angular Momentum

Theorem

Angular momentum is conserved for a body subject to no net external torque.

Proof

Let a body consist of particlesat position vectors

The motion of each particle is determined by Newton's second lawor

Taking the cross product withand summing overgives

Nowsince

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 ifthenand angular momentum is conserved.