Open Access

Robert Nicholls

• The restricted three-body problem is the study of the motion of a particle in the gravitational force field of two heavier bodies encircling their common centre of mass. This is a very old mathematical problem which goes back to Euler, Lagrange, Poincaré and others. Since those times many new theoretical fields related to dynamical systems have arrived. Often, a contact structure is required, which is in the case of the restricted three-body problem defined on an energy level set. Such contact structures have been shown to exist in the restricted three-body problem for the bounded component of the regularised energy hypersurface when the energy is below or slightly above the first critical value. This was done by Albers, Frauenfelder, van Koert and Paternain in [AFvKP12] in the planar case and Cho, Jung and Kim in [CJK20] in the spatial case. Above the energy value zero the canonical Liouville 1-form becomes a contact form. But in between these two values it was as yet unknown whetherThe restricted three-body problem is the study of the motion of a particle in the gravitational force field of two heavier bodies encircling their common centre of mass. This is a very old mathematical problem which goes back to Euler, Lagrange, Poincaré and others. Since those times many new theoretical fields related to dynamical systems have arrived. Often, a contact structure is required, which is in the case of the restricted three-body problem defined on an energy level set. Such contact structures have been shown to exist in the restricted three-body problem for the bounded component of the regularised energy hypersurface when the energy is below or slightly above the first critical value. This was done by Albers, Frauenfelder, van Koert and Paternain in [AFvKP12] in the planar case and Cho, Jung and Kim in [CJK20] in the spatial case. Above the energy value zero the canonical Liouville 1-form becomes a contact form. But in between these two values it was as yet unknown whether or not the corresponding energy level sets admit contact structures. We show in this work that the restricted three-body problem is in general not of contact type. More explicitly, sequences of generating orbits with increasingly negative action and energies between -√2 and zero are constructed. Using results from [BM00], 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 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. Here, the constructed orbits still obstruct the existence of contact structures in certain relative de Rham classes of the regularised energy hypersurface to the Liouville 1-form. Numerical results are additionally given to visualise the computations and give evidence for the existence of these orbits for higher mass ratios.…