The equations of motion for two spherical dipoles moving freely in a plane are obtained. Special consideration is given to when the two spheres are in contact. Investigations of equilibria, small-amplitude motion, and large-amplitude motion reveal that possible motions are exclusively quasi-periodic. Two distinct modes are identified, one of which is isomorphic with the simple pendulum, complete with a regime where it ceases to be periodic, and the angular displacement grows continuously at high energy.

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We analyze the rotational dynamics of six magnetic dipoles of identical strength at the vertices of a regular hexagon with a variable-strength dipole in the center. The seven dipoles spin freely about fixed axes that are perpendicular to the plane of the hexagon, with their dipole moments directed parallel to the plane. Equilibrium dipole orientations are calculated as a function of the relative strength of the central dipole. Small-amplitude perturbations about these equilibrium states are calculated in the absence of friction and are compared with analytical results in the limit of zero and infinite central dipole strength. Normal modes and frequencies are presented. Bifurcations are seen at two critical values of the central dipole strength, with bistability between these values.