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 potentials with the property that all bound orbits are also closed orbits: (1) an inverse-square central force such as the gravitational or electrostatic potential

and (2) the radial harmonic oscillator potential

The theorem was discovered by and named for the French mathematician Joseph Bertrand (1822-1900).

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 Lagrange's equations

where ${\displaystyle \omega \equiv {\frac {d\theta }{dt}}}$ and the angular momentum L = mr²ω is conserved. For illustration, the first term on the left-hand side is zero for circular orbits, and the applied inwards force ${\displaystyle {\frac {dV}{dr}}}$ equals the centripetal force requirement mrω², as expected.

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