Classical Dynamical Symmetries and Geometry of Trajectories

It is shown that all spherical symmetric potentials are capable of producing dynamical symmetries in classical one-body motions, thanks to the inevitable existence of symmetry axes associated with turning points for corresponding trajectories. This will definitely expand the class of maximally superintegrable one-body motions in central potentials that until now was considered to include only the Newtonian and Hookean cases. A simple method is proposed to identify and characterize these dynamical symmetries.