Conservation of Angular Momentum

Section 1.8 defined angular momentum as $\vec{L}=\vec{r}\times\vec{p}$. Differentiating this with respect to time (using the product rule, since both $\vec{r}$ and $\vec{p}$ can change with time): $\frac{d\vec{L}}{dt}=\frac{d\vec{r}}{dt}\times\vec{p}+\vec{r}\times\frac{d\vec{p}}{dt}$ Now $\dfrac{d\vec{r}}{dt}=\vec{v}$ and $\dfrac{d\vec{p}}{dt}=\vec{F}$ (Newton's second law), so $\frac{d\vec{L}}{dt}=\vec{v}\times\vec{p}+\vec{r}\times\vec{F}=\vec{v}\times m\vec{v}+\vec{r}\times\vec{F}$ Since $\vec{v}\times\vec{v}=0$ (any vector crossed with itself vanishes), the first term drops out entirely, leaving $\frac{d\vec{L}}{dt}=\vec{r}\times\vec{F}=\vec{\tau}$ using the definition of torque as the moment of force. So the rate of change of a system's angular momentum equals the net external torque acting on it. It follows immediately that if the net external torque is zero, $\dfrac{d\vec{L}}{dt}=0$, i.e. $\vec{L}$ stays CONSTANT -- this is the principle of conservation of angular momentum, the exact rotational analogue of the conservation of linear momentum (which likewise follows from $\dfrac{d\vec{p}}{dt}=0$ whenever the net external force vanishes). …