, Volume 12, Issue 1, pp 143-182
Date: 12 Jul 2012

Regular and Singular Reductions in the Spatial Three-Body Problem

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Abstract

The spatial three-body problem is a Hamiltonian system of nine degrees of freedom. We apply successive reductions in order to get the simplest reduced Hamiltonian, the one where all the continuous symmetries have been reduced out. After writing the Hamiltonian in Jacobi coordinates we use Deprit’s variables in order to perform the Jacobi elimination of the nodes. Then, we normalise with respect to the mean anomalies of the inner and outer ellipses in a region outside the resonance regime, and truncate the higher-order terms. The resulting system is expressed in the corresponding invariants that define the reduced space, which is a regular manifold of dimension eight. After that we use that the modulus of the total angular momentum and its projection onto the vertical axis of the inertial frame are integrals of the system. We obtain the invariants related to the symmetries generated by the two integrals and the corresponding reduced phase space, expressing the Hamiltonian in terms of these invariants. The reduced space has dimension six and is a singular space for some values of the parameters. Next, as the normalised Hamiltonian is independent of the argument of the pericentre of the outer ellipse up to first order in the small parameter, we reduce the system with respect to the symmetry related with the modulus of the angular momentum of the outer ellipse. We get the three invariants that generate the reduced two-dimensional space that may have up to three singular points. In this space we study the reduced system written in terms of these invariants analysing the relative equilibria, their stabilities and bifurcations.

To Ken Meyer with great admiration. The authors have received partial support from Project MTM 2008-03818 of the Ministry of Science and Innovation of Spain.