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Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika
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Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika, 2015, Number 4, Pages 13–24 (Mi vmumm246)  

Mathematics

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

E. A. Kudryavtseva

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
References:
Abstract: 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 $\gamma_{\omega,\varepsilon}(t)= (\mathbf{q}_{\omega,\varepsilon}(t),\mathbf{p}_{\omega,\varepsilon}(t))$ 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 $\tilde\gamma_{\omega,\varepsilon}(\tilde t):= (\omega^{2/3}\mathbf{q}_{\omega,\varepsilon}(\tilde t/\omega),\omega^{-1/3}\mathbf{p}_{\omega,\varepsilon}(\tilde t/\omega))$ 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 $\mathrm{Tr}(\tilde\gamma_{\omega,1})=4-(2\pi\omega)^2+(2\pi\omega)^3/(4\pi)+O(\omega^4)$. 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.
Received: 13.02.2013
English version:
Moscow University Mathematics Bulletin, 2015, Volume 70, Issue 4, Pages 160–170
DOI: https://doi.org/10.3103/S0027132215040026
Bibliographic databases:
Document Type: Article
UDC: 521.131, 517.925.42
Language: Russian
Citation: E. A. Kudryavtseva, “Multipliers of periodic Hill solutions in the theory of moon motion and an averaging method”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 2015, no. 4, 13–24; Moscow University Mathematics Bulletin, 70:4 (2015), 160–170
Citation in format AMSBIB
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\paper Multipliers of periodic Hill solutions in the theory of moon motion and an averaging method
\jour Vestnik Moskov. Univ. Ser.~1. Mat. Mekh.
\yr 2015
\issue 4
\pages 13--24
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\jour Moscow University Mathematics Bulletin
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\pages 160--170
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