Averaging the Planar Three-Body Problem in the Neighborhood of Double Inner Collisions

Levi–Civitaʹs regularization procedure for the two-body problem easily extends to a regularization of double inner collisions in the system consisting of two uncoupled Newtonian two-body problems. Some action-angle variables are found for this regularization, and the inner body is shown to describe ellipses on all energy levels. This allows us to define a second projection of the phase space onto the space of pairs of ellipses with fixed foci. It turns out that the initial and regularized averaged Hamiltonians of the three-body problem agree, when seen as functions on the space of pairs of ellipses. After the reduction of the problem by the symmetry of rotations, the initial and regularized averaged planar three-body problems are shown to be orbitally conjugate, up to a diffeomorphism in the parameter space consisting of the masses, the semi major axes and the angular momentum.