Quasiperiodic Motions in the Planar Three-Body Problem

Abstract

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 C k -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 Poincaré's 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 Arnold's solutions ( Russian Math. Surveys 18 (1963), 85–191) (resp. Lieberman's 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–Llibre's invariant “punctured tori” ( Ergodic Theory Dynam. Systems 8 (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 d'un Théorème de V.I. Arnold,” Séminaire de Systèmes Dynamiques and Manuscripts, 1998).