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Article Dans Une Revue Discrete and Continuous Dynamical Systems - Series B Année : 2008

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

Résumé

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) P12 family with its twelfth-order symmetry. 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 P12 follows that of Hill's orbits in the planar circular restricted three-body problem. 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.
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Dates et versions

hal-00023580 , version 1 (01-05-2006)
hal-00023580 , version 2 (23-05-2006)
hal-00023580 , version 3 (10-08-2006)
hal-00023580 , version 4 (31-10-2006)

Identifiants

  • HAL Id : hal-00023580 , version 4

Citer

Alain Chenciner, Jacques Fejoz. The flow of the equal-mass spatial 3-body problem in the neighborhood of the equilateral relative equilibrium. Discrete and Continuous Dynamical Systems - Series B, 2008, 10 (2-3), pp.421-438. ⟨hal-00023580v4⟩
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