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arXiv · 1309.1280

The Vanishing Twist in the Restricted Three Body Problem

Holger R Dullin·Joachim Worthington
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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.

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BibTeXRIS

Holger R Dullin, Joachim Worthington. 2013-09-05. The Vanishing Twist in the Restricted Three Body Problem. https://doi.org/10.1016/j.physd.2014.03.001

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