Critical points of the integral map of the charged 3-body problem

Critical points at infinity in charged N-body systems

We define the notion of critical points at infinity for the charged N-body problem, following the approach of Albouy 1993. We give a characterisation of such points and show how they can be found in the charged 3-body problem. The symmetry group of the N-body problem and accompanying integrals play a key role. In fact critical points at infinity are indispensible in understanding the bifurcations of the integral map. Together with the critical points at infinity in the charged 3-body problem, we present the bifurcation values.

Hill regions of charged three-body systems

For charged three-body systems, we discuss the configurations and orientations that are admissible for given values of the conserved total energy and angular momentum. The admissible configurations and orientations are discussed on a configuration space that is reduced by the translational, rotational and dilation symmetries of charged three-body systems. We consider the examples of the charged three-body systems given by the helium atom (two electrons and a nucleus) and the compound of two electrons and one positron. For comparison, the well known example of the Newtonian gravitational three-…

On a planar circular restricted charged three-body problem

We introduce a circular restricted charged three-body problem on the plane. In this model, the gravitational and Coulomb forces, due to the primary bodies, act on a test particle; the net force exerted by some primary body on the test particle can be attractive, repulsive or null. The restricted problem is obtained by the general planar charged three-body problem considering one mass of the three bodies going to zero. We obtain necessary restrictions for the parameters that appear in the problem, in order to be well defined. Taking into account such restriction, we study the existence and line…

Equivalence of linear stabilities of elliptic triangle solutions of the planar charged and classical three-body problems

In this paper, we prove that the linearized system of elliptic triangle homographic solution of planar charged three-body problem can be transformed to that of the elliptic equilateral triangle solution of the planar classical three-body problem. Consequently, the results of Mart\'ınez, Samà and Simó ([15] in J. Diff. Equa.) of 2006 and results of Hu, Long and Sun ([6] in Arch. Ration. Mech.Anal.) of 2014 can be applied to these solutions of the charged three-body problem to get their linear stability.

A first integral to the partially averaged Newtonian potential of the three-body problem

We consider the partial average i.e., the Lagrange average with respect to {\it just one} of the two mean anomalies, of the Newtonian part of the perturbing function in the three--body problem Hamiltonian. We prove that such a partial average exhibits a non--trivial first integral. We show that this integral is fully responsible of certain cancellations in the averaged Newtonian potential, including a property noticed by Harrington in the 60s. We also highlight its joint rôle (together with certain symmetries) in the appearance of the so called "Herman resonance". Finally, we discuss an applic…

三体 secular 理论与开普勒弹道中的共同第一积分

A common first integral from three-body secular theory and Kepler billiards

本文揭示了三体 secular 理论与开普勒弹道中的共同第一积分的联系，并定义了常曲率面上的可积弹道系统。