The dynamics around the collinear points of the elliptic three-body problem: a normal form approach (Q6599873)

The authors are concerned with the dynamics of the collinear points in the planar, restricted three-body problem, assuming that the primaries move on an elliptic orbit around a common barycenter.\N\NIn this work, the authors extend their own study of the dynamics around the collinear points in the planar \textbf{circular} restricted three-body problem [\textit{A. Celletti} et al., J. Nonlinear Sci. 25, No. 2, 343--370 (2015; Zbl 1344.70022); Physica D 317, 28--42 (2016; Zbl 1364.70022)] to the more realistic elliptic orbit. It is worth noting that the same (elliptic orbit) problem has been treated with the Poincare-Lindstedt method [\textit{X. Y. Hou} and \textit{L. Liu}, Mon. Not. R. Astron. Soc. 415, 3552--3560 (2011; \url{doi:10.1111/j.1365-2966.2011.18970.x})] by other authors.\N\NFollowing its own work [\textit{A. Celletti} et al., J. Nonlinear Sci. 25, No. 2, 343--370 (2015; Zbl 1344.70022); Physica D 317, 28--42 (2016; Zbl 1364.70022)], the authors write the equations of motion in a rotating-pulsating barycentric frame, taking the true anomaly as an independent variable. The Hamiltonian describing this problem depends on the true anomaly, and therefore it is useful to consider it in the extended phase space by adding a dummy action conjugated to the true anomaly. Thus the authors obtain a four-degree of freedom Hamiltonian. The analytic solutions for the problem are based on the construction of a resonant normal form for the collinear points \(L_1\) and \(L_2\) as in their earlier work [\textit{A. Celletti} et al., J. Nonlinear Sci. 25, No. 2, 343--370 (2015; Zbl 1344.70022); Physica D 317, 28--42 (2016; Zbl 1364.70022)]. The normal form provides an approximate solution for the Cartesian coordinates, which allows construction of several kinds of orbits, most notably planar and vertical Lyapunov orbits, as well as halo orbits.\N\NThe concrete applications concern the Earth-Moon system. In the section 6, the analytical results are compared to a numerical simulation, which requires special care in the selection of the initial conditions.