Second species orbits of negative action and contact forms in the circular restricted three-body problem

TL;DR

This paper demonstrates that in the circular restricted three-body problem, certain energy levels and negative actions prevent the existence of contact structures, with constructed orbits obstructing contact properties for small mass ratios.

Contribution

It constructs sequences of generating orbits with negative action that obstruct contact structures in the restricted three-body problem at specific energy levels and mass ratios.

Findings

Generated orbits extend to periodic solutions for small mass ratios.

Obstructions to contact structures occur at energies between -√2 and 0.

Numerical evidence supports existence of these orbits at higher mass ratios.

Abstract

We show in this work that the restricted three-body problem is in general not of contact type and zero is the energy value where the contact property breaks down. More explicitly, sequences of generating orbits with increasingly negative action and energies between $-\sqrt{2}$ and zero are constructed. Using results from Bolotin and MacKay, it is shown that these generating orbits extend to periodic solutions of the restricted three-body problem for small mass ratios and the action remains within a small neighbourhood. These orbits obstruct the existence of contact structures for energy level sets $\Sigma_c$ of the mentioned values and small mass ratios of the spatial problem. In the planar case the constructed orbits are noncontractible even in the Moser-regularised energy hypersurface $\overline{\Sigma}_c$. Here, the constructed orbits still obstruct the existence of contact structures in certain relative de Rham classes of $\overline{\Sigma}_c$ to the Liouville 1-form. These results are optimal in the sense that for energies above zero the level sets are again contact for all mass ratios. Numerical results are additionally given to visualise the computations and give evidence for the existence of these orbits for higher mass ratios.