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

Abstract

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.