The Motion of the Spherical Pendulum Subjected to a Dn Symmetric Perturbation

The motion of a spherical pendulum is characterized by the fact that all trajectories are relative periodic orbits with respect to its circle group of symmetry (invariance by rotations around the vertical axis). When the rotational symmetry is broken by some mechanical effect, more complicated, possibly chaotic behavior is expected. When, in particular, the symmetry reduces to the dihedral group Dn of symmetries of a regular n-gon, n > 2, the motion itself undergoes dramatic changes even when the amplitude of oscillations is small, which we intend to explain in this paper. Numerical simulations confirm the validity of the theory and show evidence of new interesting effects when the amplitude of the oscillations is larger (symmetric chaos).