Paper 4, Section $I$, B

The radial equation of motion of a particle moving under the influence of a central force is

$\ddot{r}-\frac{h^{2}}{r^{3}}=-k r^{n}$

where $h$ is the angular momentum per unit mass of the particle, $n$ is a constant, and $k$ is a positive constant.

Show that circular orbits with $r=a$ are possible for any positive value of $a$, and that they are stable to small perturbations that leave $h$ unchanged if $n>-3$.

*Typos? Please submit corrections to this page on GitHub.*