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

E. A. Kudryavtseva

Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 4 (2015), pp. 13-24

Résumé

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.