Rotating Eights: I. The three Γi families

We show that three families of relative periodic solutions bifurcate out of the Eight solution of the equal-mass three-body problem: the planar Hénon family, the spatial Marchal P12 family and a new spatial family. The Eight, considered as a spatial curve, is invariant under the action of the 24-element group D6 × Z2. The three families correspond to symmetry breakings where the invariance group becomes isomorphic to D6, the three D6s being embedded in the larger group in different ways. The proof of the existence of these three families relies on writing down the action integral in a rotating frame, viewing the angular velocity of the frame as a parameter, exploiting the invariance of the action under a group action which acts on the angular velocities as well as the curves and, finally, checking numerically the non-degeneracy of the Eight. Pictures and numerical evidence of the three families are presented at the end.ou