Bertrand's theorem

In classical mechanics, Bertrand's theorem states that among central-force potentials with bound orbits, there are only two types of central-force (radial) scalar potentials with the property that all bound orbits are also closed orbits. The theorem is named after its discoverer, Joseph Bertrand who published it in 1873.

The first such potential is an inverse-square central force such as the gravitational or electrostatic potential:$${\displaystyle V(r)=-{\frac {k}{r}}}$$ with force $${\displaystyle \mathbf {F} (r)=-\nabla V=-{\frac {k}{r^{2}}}{\hat {\mathbf {r} }}.}$$

The second is the radial harmonic oscillator potential:$${\displaystyle V(r)={\frac {1}{2}}kr^{2}}$$ with force $${\displaystyle \mathbf {F} (\mathbf {r} )=-\nabla V=-kr\,{\hat {\mathbf {r} }}.}$$