Ejection–Collision Orbits in Two Degrees of Freedom Problems in Celestial Mechanics

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 2D-invariant manifolds and on the behavior of the dynamics on the total collision manifold, both of them are thoroughly described.

Computer assisted proofs for transverse collision and near collision orbits in the restricted three body problem

“…These works depend on results from geometry/topology, the calculus of variations, and the KAM theory. For parameter and energy regimes where analytical results are unavailable, numerical studies illuminate the dynamics of the collision set [41,42,43,44,45,46,47,48]. Our work goes towards providing a framework for computer assisted proofs for collision orbits, for the parameter regimes where the perturbative methods cannot be applied.…”