Quasiperiodic Motions in the Planar Three-Body Problem

In the direct product of the phase and parameter spaces, we define the perturbing region, where the Hamiltonian of the planar three-body problem is Ck-close to the dynamically degenerate Hamiltonian of two uncoupled two-body problems. In this region, the secular systems are the normal forms that one gets by trying to eliminate the mean anomalies from the perturbing function. They are Pöschel-integrable on a transversally Cantor set. This construction is the starting point for proving the existence of and describing several new families of periodic or quasiperiodic orbits: short periodic orbits associated to some secular singularities, which generalize Poincaréʹs periodic orbits of the second kind (“Les Méthodes Nouvelles de la Mécanique Céleste,” Vol. 1, Gauthier-Villars, Paris, 1892–1899); quasiperiodic motions with three (resp. two) frequencies in a rotating frame of reference, which generalize Arnoldʹs solutions (Russian Math. Surveys18 (1963), 85–191) (resp. Liebermanʹs solutions; Celestial Mech.3 (1971), 408–426); and three-frequency quasiperiodic motions along which the two inner bodies get arbitrarily close to one another an infinite number of times, generalizing the Chenciner–Llibreʹs invariant “punctured tori” (Ergodic Theory Dynam. Systems8 (1988), 63–72). The proof relies on a sophisticated version of KAM theorem, which itself is proved using a normal form theorem of Herman (“Démonstration dʹun Théorème de V.I. Arnold,” Séminaire de Systèmes Dynamiques and Manuscripts, 1998).