September  2008, 10(2&3, September): 421-438. doi: 10.3934/dcdsb.2008.10.421

The flow of the equal-mass spatial 3-body problem in the neighborhood of the equilateral relative equilibrium

1. 

ASD, IMCCE (UMR 8028), Observatoire de Paris, 77 avenue Denfert-Rochereau, 75014 Paris, France

2. 

Université P. & M. Curie (Paris VI), Institut de Mathématiques (UMR 7586), Analyse algébrique, 175 rue du Chevaleret, 75013 Paris, France

Received  October 2006 Revised  June 2007 Published  June 2008

From a normal form analysis near the Lagrange equilateral relative equilibrium, we deduce that, up to the action of similarities and time shifts, the only relative periodic solutions which bifurcate from this solution are the (planar) homographic family and the (spatial) $P_{12}$ family with its twelfth-order symmetry (see [13, 5]). After reduction by the rotation symmetry of the Lagrange solution and restriction to a center manifold, our proof of the local existence and uniqueness of $P_{12}$ follows that of Hill's orbits in the planar circular restricted three-body problem in [7, 1]. Indeed, near the Lagrange solution, the restrictions of constant energy levels of the reduced flow to a center manifold (actually unique) turn out to be three-spheres. In an annulus of section bounded by relative periodic solutions of each family, the normal resonance along the homographic family entails that the Poincaré return map is the identity on the corresponding connected component of the boundary. Using the reflexion symmetry with respect to the plane of the relative equilibrium, we prove that, close enough to the Lagrange solution, the return map is a monotone twist map.
Citation: Alain Chenciner, Jacques Féjoz. The flow of the equal-mass spatial 3-body problem in the neighborhood of the equilateral relative equilibrium. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 421-438. doi: 10.3934/dcdsb.2008.10.421
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