Quasiperiodic motions in the planar three-body problem
The paper discusses the secular dynamics and the existence of new families of periodic and quasiperiodic orbits in the planar three-body problem. Firstly, by introducing the Jacobi coordinates, the author represents the Hamiltonian of the problem in the form \(F=F_{\text{Kep}}+F_{\text{per}}\), where \(F_{\text{Kep}}\) describes the direct product of Keplerian motions of two fictitious bodies revolving along inner and outer ellipses around a fixed centre of attraction, without mutual interaction. The perturbation \(F_{\text{per}}\) is real analytic outside the collisions, and is \(\varepsilon\)-small in the \(C^k\)-norm in the so-called perturbing region which is a special subset of the direct product of phase and parameter spaces. It is possible to eliminate the fast angles from \(F\), up to same order, by a suitably chosen \(C^{\infty}\)-symplectomorphism \(\phi^n\), so that \(\phi^{n*}F\) splits into the sum of the secular Hamiltonian \(F_{\pi}^n\) and of the complementary part \(F_{\text{comp}}^n\) of the order \(O(\varepsilon^{1+n})\). In turn, the secular term can be split into two parts: \(F_{\pi}^n=F_{\text{int}}^n+F^n_{\text{res}}\). Depending on the choice of the perturbing region, the infinite jet of the resonant part \( F^n_{ \text{res} }\) vanishes either on a transversal Cantor set (defined by frequency vectors satisfying certain diophantine conditions) or everywhere, so that one can deal with completely integrable part \(F_{\text{int}}^n\) only. Applying KAM techniques described in [\textit{M. R. Herman}, Démonstration d'un théorème de V. I. Arnold, Seminaire de Systèmes dynamiques and manuscript, Univ. Paris-Diderot (1998)], the author proves the existence of a positive measure of quasiperiodic Lagrangian tori in the dynamics of the planar-three body problem inside the perturbing region. The solutions obtained in the paper generalize the Poincaré periodic orbits of the second kind, the Arnold's solutions and the Chenciner-Llibre's invariant ``punctured tori''.
scientific article; zbMATH DE number 1836798