 Solution

In a short time $dt$, the radius vector sweeps an oriented area $d\mathcal{A}=\frac{1}{2}{r}^{2}d\theta$. ThusThe constant $h$ is twice the oriented areal velocity, or the angular momentum per unit mass. Its constancy is Kepler's second law: equal areas are swept in equal times.

For a circular orbit of radius $a$, set $\ddot{r}=0$ in the radial equation. The balance isHence a circular orbit exists for any $a>0$ with this nonzero $|h|$. Its period gives Kepler's third law:To check circular-orbit stability for a power-law central force directly, keep $h$ fixed and set $r=a+\rho$. Expandingabout the balanced radius givesThe linear radial perturbation therefore obeysIt remains bounded and oscillates, proving stability under the stipulated small perturbations. Equivalently, the specific effective potential $h^2/(2r^2)-GM/r$ has a strict minimum at $a$.