Multipliers of periodic Hill solutions in the theory of moon motion and an averaging method

A 2-parametric family of Hamiltonian systems H ω,e with two degrees of freedom is studied, where the system H ω,0 describes the Kepler problem in rotating axes with the angular frequency ω, and the system H 1, 1 describes the Hill problem, i.e. a “limiting” motion of the Moon in the planar three body problem “Sun-Earth-Moon” with the masses m 1 ≫ m 2 > m 3 = 0. Using an averaging method on a submanifold, we prove the existence of a number ω0 > 0 and a smooth family of 2π-periodic solutions γ ω,e (t) = (q ω,e (t),p ω,e (t)) to the system H ω,e |e| ≤ 1 |ω| ≤ ω0, such that γ ω ,0 are circular solutions, and γ ω,e = γ ω_,0 + O(ω 2 e) together with the “rescaled” motions γ ω,e (\(\widetilde t\)): = (ω 2/3 q ω,e (\(\widetilde t\)/ω),ω -1/3 p ω,e (\(\widetilde t\)/ω)) for 0 < |ω| ≤ ω0 and e= 1 form two families of Hill solutions, i.e., the initial segments of the known families f and g+(with a reverse and direct directions of the motion) of 2πω-periodic solutions to the Hill problem H 1, 1. Using averaging, it is also proved that the sum of multipliers of the Hill solution γ ω,1 has the form Tr (γ ω,1) = 4 — (2πω)2 + (2πω)3/(4π) + O(ω 4). The results are refined and extended to a certain class of systems including the restricted three body problem, as well as applied to planetary systems with satellites.