Dynamics in the center manifold of the collinear points of the restricted three body problem

Abstract

This paper focuses on the dynamics near the collinear equilibrium points L 1,2,3 of the spatial Restricted Three Body Problem (RTBP). It is well known that the linear behavior of these three points is of the type center×center×saddle. To obtain an accurate description of the dynamics in an extended neighborhood of those points, two different (but complementary) strategies are used. First, the Hamiltonian of the problem is expanded in power series around the equilibrium point. Then, a partial normal form scheme is applied in order to uncouple (up to high order) the hyperbolic directions from the elliptic ones. Skipping the remainder we have that the (truncated) Hamiltonian has an invariant manifold tangent to the central directions of the linear part. The restriction of the Hamiltonian to this manifold is the so-called reduction to the center manifold. The study of the dynamics of this reduced Hamiltonian (now with only two degrees of freedom) gives a qualitative description of the phase space near the equilibrium point. Finally, a Lindstedt–Poincaré procedure is applied to explicitly compute the invariant tori contained in the center manifold. These tori are obtained as the Fourier series of the corresponding solutions, the frequencies being a power expansion of some parameters (amplitudes). This allows for an accurate quantitative description of these regions. In particular, the well known Halo orbits are obtained.