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 first such potential is an inverse-square central force such as the gravitational or electrostatic potential:

$${\displaystyle V(r)=-{\frac {k}{r}}}$$ with force $${\displaystyle f(r)=-{\frac {dV}{dr}}=-{\frac {k}{r^{2}}}.}$$

The second is the radial harmonic oscillator potential:

$${\displaystyle V(r)={\frac {1}{2}}kr^{2}}$$ with force $${\displaystyle f(r)=-{\frac {dV}{dr}}=-kr.}$$

The theorem is named after its discoverer, Joseph Bertrand.

All attractive central forces can produce circular orbits, which are naturally closed orbits. The only requirement is that the central force exactly equals the centripetal force, which determines the required angular velocity for a given circular radius. Non-central forces (i.e., those that depend on the angular variables as well as the radius) are ignored here, since they do not produce circular orbits in general.

The equation of motion for the radius ${\displaystyle r}$ of a particle of mass ${\displaystyle m}$ moving in a central potential ${\displaystyle V(r)}$ is given by motion equations