Quasiperiodic motions in the planar three-body problem

[摘要] 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 Poschel-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 Poincare's periodic orbits of the second kind (Les Methodes Nouvelles de la Mecanique Celeste, 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 (Demonstration d'un Theoreme de V.I. Arnold, Seminaire de Systemes Dynamiques and Manuscripts, 1998). (C) 2002 Elsevier Science (USA).

[发布日期] 2002-08-10

[关键词] three-body problem;secular system;averaging;regularization;KAM theorem;periodic orbits