Bi-normal trajectories in the Circular Restricted Three-Body Problem

TL;DR

This paper demonstrates the existence of infinitely many bi-normal trajectories in the spatial circular restricted three-body problem near the primaries, using advanced topological methods under specific energy conditions.

Contribution

It applies the relative Poincaré-Birkhoff theorem to establish the existence of bi-normal trajectories in a complex dynamical system, extending previous theoretical results.

Findings

Existence of infinitely many bi-normal trajectories near primaries

Trajectories are normal at initial and final times

Results hold for energies below or slightly above the first critical value

Abstract

In this note, we show there exist infinitely many trajectories which are bi-normal (i.e. normal at initial and final times) to the xz-plane, in the Spatial Circular Restricted Three-Body Problem, for energies below or slightly above the first critical value and near the primaries, under the assumption of the twist condition as defined by Moreno-van-Koert in arXiv:2011.06562. This is an application of the relative Poincar\'e-Birkhoff theorem for Lagrangians in Liouville domains, as proven by the authors in arXiv:2408.06919.