Past exam of the mathematics course of the University of Cambridge / 2009 / ia / Paper 4 / 4A / b / Solution

The particle's angular momentum about the force centre is $\mathbf{L}=m\mathbf{r}\times \dot{\mathbf{r}}$. A central force exerts zero torque about that centre, soBy conservation of angular momentum, $\mathbf{L}$ is a constant vector. If it is nonzero, the identity $\mathbf{r}\cdot \mathbf{L}=0$ places every point of the orbit in the fixed plane through the centre perpendicular to $\mathbf{L}$:If $\mathbf{L}=0$, position and velocity are parallel. Away from the force centre, ${\dot{\mathbf{e}}}_r=(\dot{\mathbf{r}}-\dot{r}{\mathbf{e}}_r)/r=0$, so the motion is radial along a fixed line, which lies in any plane containing that line. Thus the zero-angular-momentum case also gives planar motion. As usual, these assertions apply while the force law supplies a regular orbit; continuation through a singular force centre is a separate issue.