Ejection-collision orbits in two degrees of freedom problems in celestial mechanics

TL;DR

This paper proves the existence of ejection-collision orbits in two degrees of freedom Hamiltonian systems modeling celestial mechanics, focusing on singularities related to total and partial collisions.

Contribution

It establishes the existence of heteroclinic ejection-collision orbits connecting total collision states in a generalized celestial mechanics model.

Findings

Existence of heteroclinic orbits connecting total collision points.

Analysis of invariant manifolds and their transversality.

Characterization of partial collision trajectories.

Abstract

In a general setting of a Hamiltonian system with two degrees of freedom and assuming some properties for the undergoing potential, we study the dynamics close and tending to a singularity of the system which in models of $N$-body problems corresponds to total collision. We restrict to potentials that exhibit two more singularities that can be regarded as two kind of partial collisions when not all the bodies are involved. Regularizing the singularities, the total collision transforms into a 2-dimensional invariant manifold. The goal of this paper is to prove the existence of different types of ejection-collision orbits, that is, orbits that start and end at total collision. Such orbits are regarded as heteroclinic connections between two equilibrium points and are mainly characterized by the partial collisions that the trajectories find on their way. The proof of their existence is based on the transversality of 2-dimensional invariant manifolds and on the behavior of the dynamics on the total collision manifold, both of them are thoroughly described.