We consider a Kepler problem in dimension two or three, with a timedependent T -periodic perturbation. We prove that for any prescribed positive integer N , there exist at least N periodic solutions (with period T ) as long as the perturbation is small enough. Here the solutions are understood in a general sense as they can have collisions. The concept of generalized solutions is defined intrinsically and it coincides with the notion obtained in Celestial Mechanics via the theory of regularization of collisions.

In this article, we investigate the mathematical part of De Sitter's theory on the Galilean satellites, and further extend this theory by showing the existence of some quasi-periodic librating orbits by applications of KAM theorems. After showing the existence of De Sitter's family of linearly stable periodic orbits in the Jupiter-Io-Europa-Ganymede model by averaging and reduction techniques in the Hamiltonian framework, we further discuss the possible extension of this theory to include a fourth satellite Callisto, and establish the existence of a set of positive measure of quasi-periodic librating orbits in both models for almost all choices of masses among which one sufficiently dominates the others.

In this article, we first present the Kustaanheimo-Stiefel regularization of the spatial Kepler problem in a symplectic and quaternionic approach. We then establish a set of action-angle coordinates, the so-called LCF coordinates, of the Kustaanheimo-Stiefel regularized Kepler problem, which is consequently used to obtain a conjugacy relation between the integrable approximating "quadrupolar" system of the lunar spatial three-body problem and its regularized counterpart. This result justifies the study of of the quadrupolar dynamics of the lunar spatial three-body problem near degenerate inner ellipses.Date: October 30, 2018.

In a system of particles, quasi‐periodic almost‐collision orbits are collisionless orbits along which two bodies become arbitrarily close to each other—the lower limit of their distance is zero but the upper limit is strictly positive—and are quasi‐periodic in a regularized system up to a change of time. Their existence was shown in the restricted planar circular three‐body problem by A.~Chenciner and J. Llibre, and in the planar three‐body problem by J. Féjoz. In the spatial three‐body problem, the existence of a set of positive measure of such orbits was predicted by C. Marchal. In this article, we present a proof of this fact.© 2015 Wiley Periodicals, Inc.

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The aim of this note is to explain the integrability of an integrable Boltzmann billiard model, previously established by Gallavotti and Jauslin [G. Gallavotti and I. Jauslin, A theorem on Ellipses, an integrable system and a theorem of Boltzmann, preprint (2020); arXiv:2008.01955], alternatively via the viewpoint of projective dynamics. We show that the energy of a corresponding spherical problem leads to an additional first integral of the system equivalent to Gallavotti–Jauslin’s first integral. The approach also leads to a family of integrable billiard models in the plane and on the sphere defined through the planar and spherical Kepler–Coulomb problems.

Abstract. By application of KAM theorem to Lidov-Ziglin's global study of the quadrupolar approximation of the spatial lunar three-body problem, we establish the existence of several families of quasi-periodic orbits in the spatial lunar three-body problem.

Abstract:In [ ], Ortega has analyzed "generalized" collision solutions of the periodically forced rectilinear Kepler problem. In this note, we explain a di erent approach to study these solutions by embedding the nonautonomous Hamiltonian system into the zero-energy level of an autonomous Hamiltonian system and by employing the Levi-Civita regularization to regularize the double collisions. In addition to this, under a certain smoothness hypothesis of the periodic force term, we show that there exists a set of positive measure of generalized quasi-periodic solutions in the extended phase space, each of them accumulated by generalized periodic solutions of the system. The energy of these quasi-periodic solutions can have an arbitrary large absolute value.