Multiplicity of forced oscillations for the spherical pendulum acted on by a retarded periodic force

We prove a multiplicity result for forced oscillations of a spherical pendulum (that is, a massive point moving on a sphere) subject to a periodic action, with or without friction, allowed to depend on the whole past of the motion. The approach is based on topological methods. In particular, when the unperturbed forcing term is the gravity, we obtain two harmonic forced oscillations regardless of the presence of friction and of the form of the perturbing force field.