On closed but non-geometrically similar orbits

As is well known, the existence of geometrically similar orbits for a particle moving under a central conservative force is a consequence of the fact that the corresponding potential energy is a homogeneous function of the co-ordinates. In this paper, we consider a particular non-homogeneous potential of the form V = U + W, where U and W are homogenous functions of degrees −1 and −2, respectively, because, for this potential, the search of closed orbits, for discrete values of the angular momentum, is straightforward. We focus our attention on these daisy-like charming orbits and graphically show the consequences of the impossibility of geometrical similarity.