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

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.