Bertrand's theorem

Remarks

Both cases occur: for the Kepler problem the bounded orbits are ellipses with a focus at $0$, and for the isotropic harmonic oscillator they are ellipses centred at $0$. Idea of the proof: by conservation of angular momentum the motion is planar, and the radius oscillates in the effective potential $U(r) + \frac{\ell^2}{2 r^2}$, where $\ell$ is the angular momentum. The orbit closes exactly when the angle swept between two successive pericentres is a rational multiple of $2 \pi$. For orbits close to a stable circular orbit of radius $\rho$, this angle is $2 \pi / \sqrt{3 + \rho \, U''(\rho) / U'(\rho)}$. By continuity it must be the same rational multiple of $2 \pi$ for all such $\rho$. This forces $U'(r)$ to be a power $r^{\beta^2 - 3}$ times a constant, and a study of orbits with large amplitude leaves only $\beta = 1$ and $\beta = 2$.

The assumption that bounded orbits exist that are not circles is necessary. For example, take the repulsive potential $U(r) = 1/r$. The second derivative of $\norm{x}^2$ in time is $2 \norm{x'}^2 + 2 / \norm{x}$. Along a motion with $\norm{x} \le R$ it would stay above $2/R$, which is impossible for all $t \in \R$, so no motion is bounded. So the second assumption holds, but $U$ is not of the two forms with $k > 0$.

Going deeper

For a student

For any other central force, a typical bounded orbit is a rosette whose pericentre turns a little at each revolution and never closes. So closed planetary orbits are a strong test of the inverse-square law. The small observed precession of the orbit of Mercury shows that the Newtonian force of the Sun is perturbed, by the other planets and by general relativity. In the Kepler problem the closedness of orbits is explained by the conservation of the Laplace–Runge–Lenz vector, and in the oscillator by the separation into two oscillations with equal frequency.