Rotating Eights: I. The three Gammai families

Alain Chenciner (1, 2, 3) , Jacques Féjoz (1, 2, 4) , Richard Montgomery (5)

1 IMCCE - Institut de Mécanique Céleste et de Calcul des Ephémérides
2 ASD - Astronomie et systèmes dynamiques
3 UFR Mathématiques UPCité - UFR Mathématiques [Sciences] - Université Paris Cité
4 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
5 Department of Mathematics, University of California

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.

Domaines: Astrophysique [astro-ph]

Alain Chenciner, Jacques Féjoz, Richard Montgomery. Rotating Eights: I. The three Gammai families. Nonlinearity, 2005, 18, pp.1407-1424. ⟨10.1088/0951-7715/18/3/024⟩. ⟨hal-03742601⟩