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: with force . The second is the radial harmonic oscillator potential: with force . 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 r of a particle of mass m moving in a central potential V(r) is given by motion equations where , and the angular momentum L = mr2ω is conserved. For illustration, the first term on the left is zero for circular orbits, and the applied inwards force equals the centripetal force requirement mrω2, as expected. The definition of angular momentum allows a change of independent variable from t to θ: giving the new equation of motion that is independent of time: This equation becomes quasilinear on making the change of variables and multiplying both sides by (see also Binet equation): As noted above, all central forces can produce circular orbits given an appropriate initial velocity. However, if some radial velocity is introduced, these orbits need not be stable (i.e., remain in orbit indefinitely) nor closed (repeatedly returning to exactly the same path). Here we show that a necessary condition for stable, exactly closed non-circular orbits is an inverse-square force or radial harmonic oscillator potential.