The Vanishing Twist in the Restricted Three Body Problem

TL;DR

This paper investigates the existence of twistless tori and bifurcations in the planar circular restricted three-body problem near Lagrangian point L4, combining numerical integration with Birkhoff normal form analysis.

Contribution

It demonstrates the presence of twistless tori and reconnection bifurcations in the restricted three-body problem using numerical and analytical methods, providing a comprehensive overview of their behavior.

Findings

Twistless tori are found near L4 and associated bifurcations.

Numerical rotation numbers match Birkhoff normal form predictions.

Global action space analysis illustrates bifurcation scenarios.

Abstract

This paper demonstrates the existence of twistless tori and the associated reconnection bifurcations and meandering curves in the planar circular restricted three-body problem. Near the Lagrangian equilibrium $\mathcal{L}_4$ a twistless torus is created near the tripling bifurcation of the short period family. Decreasing the mass ratio leads to twistless bifurcations which are particularly prominent for rotation numbers 3/10 and 2/7. This scenario is studied by numerically integrating the regularised Hamiltonian flow, and finding rotation numbers of invariant curves in a two-dimensional Poincar\'{e} map. To corroborate the numerical results the Birkhoff normal form at $\mathcal{L}_4$ is calculated to eighth order. Truncating at this order gives an integrable system, and the rotation numbers obtained from the Birkhoff normal form agree well with the numerical results. A global overview for the mass ratio $\mu \in (\mu_4, \mu_3)$ is presented by showing lines of constant energy and constant rotation number in action space.