New Bound Closed Orbits in Spherical Potentials

We present new families of bound, closed, nonelliptical orbits that are supported by various spherical potentials in clear contradiction to Newton’s and Bertrand’s theorems. We calculate analytically some typical closed orbits of arbitrarily large amplitudes for the spherically symmetric forms of the Newton-Kepler, Hooke, and linear potentials. In all cases, the radial oscillations of the orbits exhibit unequal amplitudes and nonsynchronous frequencies on either side of a circular equilibrium orbit. These parameters are however related through energy conservation and orbit continuity across the equilibrium orbit. The constraints placed on the parameters are all localized to each particular orbit and depend crucially on the chosen initial conditions.