Dynamics of the Parabolic Restricted Collinear Three-Body Problem

Abstract

We give a complete study of the dynamics of an infinitesimal mass under the Newtonian attraction of two point masses---particles which escape along a line with zero energy---and a third massless particle moving along the same line (called the parabolic restricted collinear three-body problem). The dynamics is governed by a time-dependent Hamiltonian formulation which is dissipative, and we must study separately the interior case (the massless particle is between the primaries) and the exterior case (the massless particle is to the right or left of the primaries). Regularization of binary collisions is achieved using Birkhoff transformation, which must be performed in the full planar restricted parabolic three-body problem using complex variables. We prove that the system in pulsating coordinates is a gradient-like dynamical system embedded in $S^1\times\mathbb{R}\times\mathbb{R}$ or $\mathbb{R}^+\times\mathbb{R}\times\mathbb{R}$ with a first integral. We study the dynamics on the energy surface together with the dynamics at infinity using Poincaré compactification. Using the stable and the unstable branches, we are able to describe analytically the global dynamics, and to prove the existence of different motions, in particular the existence of parabolic and hyperbolic solutions.