Angular momentum

The angular momentum $\mathbf{L}$ of an object at position $\mathbf{r}$ moving about a point $O$ with linear momentum $\mathbf{p}$ is the cross product

$\mathbf{r}$ is the vector connecting $O$ with the object.

A moment of force $\mathbf{N}$ applied onto the object changes the angular momentum of the object according to:

also using dot notation for the derivative. From this we can immediately see the very important theorem of angular momentum conservation.

An equivalent theorem also holds for a system of particles:

The clause about the direction of the internal forces is necessary to cancel out the internal moments of force, as they read $\mathbf{r}_{ij}\times \mathbf{F}_{ij}$, with $i$ and $j$ the indexes of individual particles. If $\mathbf{r}_{ij}$ and $\mathbf{F}_{ij}$ are not parallel, these do not cancel and the theorem does not hold. External moments of force are those applied to the particles by external sources and not other particles.