Multiplicators of periodic Hill solutions in the theory of Moon motion and the method of averaging / E. A. Kudryavtseva. // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2015. № 4. P. 13-24
[Moscow Univ. Math. Bulletin. Vol. 72, N 2, 2017. P. 160-170].

A 2-parameter family of Hamiltonian systems \mathcal H_{\omega,\varepsilon} with two degrees of freedom
is studied, where the system \mathcal H_{\omega,0} describes the Kepler problem in rotating axes with angular
frequence \omega, the system \mathcal 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\gg m_2>m_3=0.
Using the averaging method on a submanifold, we prove the existence of \omega_0>0 and a smooth
family of 2\pi-periodic solutions \gamma_{\omega,\varepsilon}(t)=
({\bf q}_{\omega,\varepsilon}(t),{\bf p}_{\omega,\varepsilon}(t)) to the system
\mathcal H_{\omega,\varepsilon}, |\varepsilon|\le1, |\omega|\le\omega_0, such that
\gamma_{\omega,0} are cirlular solutions, \gamma_{\omega,\varepsilon}=\gamma_{\omega,0}+O(\omega^2\varepsilon),
and the "rescaled" motions \tilde\gamma_{\omega,\varepsilon}(\tilde t):=
(\omega^{2/3}{\bf q}_{\omega,\varepsilon}(\tilde t/\omega),\omega^{-1/3}{\bf p}_{\omega,\varepsilon}(\tilde t/\omega))
for 0<|\omega|\le\omega_0 and \varepsilon=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 motion)
of 2\pi\omega-periodic solutions of the Hill problem \mathcal H_{1,1}. Using averaging, we prove that the sum of
the multipliers of the Hill solution \tilde\gamma_{\omega,1} has the form
{\rm Tr}\,(\tilde\gamma_{\omega,1})=4-(2\pi\omega)^2+(2\pi\omega)^3/(4\pi)+O(\omega^4).
The results are developed and extended to a class of systems including the restricted three body problem,
as well as applied to planetary systems with satellites.

Key words:
three body problem, Hill problem, periodic solutions, averaging on a submanifold.