We show that the planar circular restricted three-body problem is of restricted contact type for all energies below the first critical value (action of the first Lagrange point) and for energies slightly above it. This opens up the possibility of using the technology of contact topology to understand this particular dynamical system.
In this article we explain how critical points of a particular perturbation of the Rabinowitz action functional give rise to leaf-wise intersection points in hypersurfaces of restricted contact type. This is used to derive existence and multiplicity results for leaf-wise intersection points in hypersurfaces of restricted contact type in general exact symplectic manifolds. The notion of leaf-wise intersection points was introduced by Moser [Mos78].2000 Mathematics Subject Classification. 53D40, 37J10, 58J05. Key words and phrases. Leaf-wise Intersections, Rabinowitz Floer homology, stretching the neck, local homology.
A perturbation of the Rabinowitz action functionalWe recall that Σ ⊂ (M, ω = dλ) is a closed hypersurface in an exact symplectic manifold such that (Σ, α = λ |Σ ) is a contact manifold. Moreover, Σ is assumed to bound a compact region in M . We denote by R the Reeb vector field of α. Moreover, we define the vector field Y by dλ(Y, ·) = λ(·).Lemma 2.1. The vector field Y is a Liouville vector field for (Σ, α), that is, L Y ω = ω and Y ⋔ Σ. In particular, (Σ, α) is of restricted contact type.
The restricted planar three-body problem has a rich history, yet many unanswered questions still remain. In the present paper we prove the existence of a global surface of section near the smaller body in a new range of energies and mass ratios for which the Hill's region still has three connected components. The approach relies on recent global methods in symplectic geometry and contrasts sharply with the perturbative methods used until now.
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