Seminars ASD

On s'intéresse au contrôle de l'équation de Kepler afin de modéliser la trajectoire d'un engin spatial que l'on souhaite transférer d'une orbite périodique vers une orbite périodique, dans le plan. Ce problème se plonge dans une famille à deux paramètres dont l'un représente la masse d'un troisième corps (problème circulaire restreint contrôlé), l'autre le module du contrôle exercé. En l'absence de contrôle et de masse additionnelle, le problème est classiquement intégrable par quadratures alors qu'il existe des obstructions dès que la troisième masse est non nulle. Dans le cas de deux corps, le problème contrôlé pour lequel on cherche à minimiser la norme L^{^2} du contrôle possède un moyennisé dont les trajectoires sont géodésiques et intégrables. Le but de cet exposé est de montrer que la minimisation du temps pour le problème de Kepler donne lieu a des obstructions de nature Galois différentielles à l'intégrabilité.

Travaux en commun avec  T. Combot, J. Féjoz et M. Orieux, publiés dans J. Geom. Phys. 132 (2018), 452-459.

Minimizing methods have been successfully applied to construct various types of periodic solutions for the n-body and n-center problems during the past two decades. Majority of relevant researches were endeavored to understand qualitative features such as existence, uniqueness, and stability. In this talk we discuss a topic with relatively less attention — quantitative estimates for action values and mutual distances for action minimizing solutions. We will demonstrate some simple but nontrivial bounds. These estimates will facilitate numerical explorations to effectively locate and search new orbits.

Extrasolar planetary systems commonly exhibit planets on eccentric orbits, with many systems located near or within mean-motion resonances, showcasing a wide diversity of orbital architectures. Such complex systems challenge traditional secular theories, which are limited to first-order approximations in planetary masses or rely on expansions in orbital elements—eccentricities, inclinations, and semi-major axis ratios—that are subject to convergence issues, especially in highly eccentric, inclined, or tightly-packed systems. In this talk, we present a numerical approach to second-order perturbation theory, developed using the Lie series formalism, to address these limitations. We first outline the Hamiltonian framework for the three-body planetary problem and apply a canonical transformation to eliminate fast angle dependencies, deriving the secular Hamiltonian up to second order in the mass ratio. We then use the fast Fourier transform algorithm to numerically simulate, with high precision, the long-term evolution of planetary systems near or away from mean-motion resonances. Finally, we validate our methods against well-known planetary configurations, such as the Sun–Jupiter–Saturn system, as well as exoplanetary systems like WASP-148, TIC 279401253, and GJ 876, demonstrating the applicability of our model across a wide range of orbital architectures.

Polynomial systems of equations and inequalities encode naturally non-linear geometric and arithmetic (static) properties. Consequently, they arise in many scientific fields. However, because of their non-linearity, they are difficult to handle through purely numerical methods, especially when the end-user expects to compute global information on the solution set.

In this talk, we will review algebraic methods for solving such systems,
commenting on their strengths and weaknesses with an emphasis on geometric applications and algorithms for solving over the real numbers.

The n-body problem of celestial mechanics has simple motions called the relative equilibria. The configuration of the bodies in such a motion is called central.

In 1996 I proved by using polynomial elimination on Maple that there are only 3 types of symmetric non-collinear central configurations of 4 equal masses. My preceding paper had proved the symmetry of these planar central configurations.

In 2019 another computer assisted method, based on interval arithmetic, reproved these results and the similar ones for 5, 6 and 7 bodies with equal masses (Moczurad, Zgliczynski).

These methods are less efficient when dealing with equations with parameters. Here the important parameters are the masses of the bodies.
Faugère, Kotsireas and Lazard studied in particular the symmetric case with 4 bodies, using polynomial elimination, showing a lot of bifurcations lines where the number of solutions changes, many of them out of the domain of positive masses. The thesis of Leandro (2003) under the direction of Moeckel gave a simple picture. A recent publication by Roberts (2025) reconsiders the question. I will try to present these works.

We consider the planar circular restricted three-body problem, modeling the motion of a massless asteroid in the plane undergoing gravitational attraction toward two bodies, each of which moves in a circular path around their common center of mass. For small mass ratios, the motion of the asteroid is approximated by the Kepler system when the asteroid is far from a collision, leading to quasi-periodic motions in the region of the phase space where the paths traced by the asteroid and the smaller body do not intersect. However, in the region of the phase space where the paths do intersect, the potential for close interactions between the asteroid and the smaller body destroy these quasi-periodic motions. The existence of hyperbolic sets in which the asteroid repeatedly comes close to a collision was proven independently by Bolotin and MacKay and by Font, Nunes, and Simó. My result, currently in preparation, is that there also exist elliptic periodic orbits near Kepler orbits with resonant frequencies in which the asteroid repeatedly undergoes close interactions with the smaller body. Furthermore, these elliptic periodic orbits are surrounded by elliptic islands, and the total measure of the elliptic islands in the phase space is bounded below by a positive constant.