Angular Momentum

Definition

The motion of a material point (with unit mass) in a central field on a plane is defined by the equation $\ddot{\mathbf{r}}=\Phi (r){\mathbf{e}}_r,$ where $\mathbf{r}$ is the radius vector beginning at the center of the field $0$, $r$ is its length, and ${\mathbf{e}}_r$ its direction. We will think of our plane as lying in three-dimensional oriented euclidean space.

Definition

The angular momentum of a material point of unit mass relative to the point $0$ is the vector product $\mathbf{M}=[\mathbf{r},\dot{\mathbf{r}}]=\mathbf{r}\times \dot{\mathbf{r}}$ The vector $\mathbf{M}$ is perpendicular to our plane and is given by one number: $\mathbf{M}=M\mathbf{n}$, where $\mathbf{n}=[{\mathbf{e}}_1,{\mathbf{e}}_2]$ is the normal vector, ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$ being an oriented frame in the plane.

The law of conservation of angular momentum

Under motions in a central field, the angular momentum $\mathbf{M}$ relative to the center of the field $0$ does not change with time.