Closed orbits in central forces distinct from Coulomb or harmonic oscillator type

Particles moving in a central force move in closed nonprecessing orbits for discrete values of the angular momentum even though the force law is different from the inverse square (Coulomb) or linear (harmonic oscillator) type. We do not present a general survey of all central forces exhibiting such discrete closed orbits because the search is likely to be tedious. We restrict ourselves to the family of central force laws for which it is always possible to find a function of u , the inverse radial distance 1/ r , which regarded as a function of the angular coordinate ( is the orbit equation) is a harmonic function for all initial conditions. Surprisingly for all central forces of this family the search of closed orbits is straightforward and the discrete values of the angular momentum J are easily found.