Search for periodic solutions with special symmetry in the hill problem

Hill problem is a well-known celestial mechanics problem, which gives the description of dynamics of a satellite near the minor of two active masses in the uniformly rotating frame (sinodical system of coordinates). This problem is a limiting case of the well-known restricted three-body problem (RTBP), and its periodical solutions can be continued into the corresponding solutions of the RTBP. We propose to embed the Hill problem in so-called generalized Hill problem, which Hamiltonian can be written in the form = 𝐻0 + "𝑅, where 𝐻0 = 1 2 (︀ 𝑦2 1 + 𝑦2 2 )︀ + 𝑥2𝑦1 - 𝑥1𝑦2 + , = -𝑥21 + 1 2 𝑥22 , = √︁ 𝑥21 + 𝑥22 . Here and are vectors of canonically conjugate coordinates and momenta correspondingly, " ∈ [0; 1] and ∈ {-1, 0} are some parameters. Canonical equations of motions of the problem are invariant under the discrete group with two generators of linear transformations of the extended phase space. Therefore all periodic solutions drop into the four group of asymmetric, singly symmetric, doubly symmetric and centrally symmetric solutions. Singly and doubly symmetric periodic solutions were intensively studied with the help of singular perturbation theory earlier. But some heuristics make it possible to state that centrally symmetric orbits exist. In this paper we compute generating solutions with central symmetry of the Hill problem. The method of computing centrally symmetric generating solutions is based on iterative calculation of a normal form of the Hill problem Hamiltonian in the vicinity of periodic solutions of the Kepler problem in the uniformly rotating frame. The method of invariant normalization (symmetrization) proposed by academician V. F. Zhuravlev was applied. The method is based on integration of so called homological equation for obtaining both Lie generator of normal transformation and normal form ℋ. The normal form ℋ2 of the second order of initial Hamiltonian was computed and condition on existence of centrally symmetric generating solutions was formulated. Their initial conditions and periods were computed as well. These generating solutions can be used for their numerical continuation along the parameter " up to periodic solutions with central symmetry of the Hill problem.