Workshop in Celestial Mechanics in honor of Alain Albouy
Paris, July 7-9 2025

A point mass moves inside a convex set \(\Omega\subset \mathbb{R}^2\) under the gravitational attraction of a positive fixed mass. When it hits the boundary it gets reflected back inside \(\Omega\) according to the geometric optics rule \emph{the angle of incidence equals the angle of reflection}. It is know that this system is integrable (for all energies) when \(\Omega\) is an ellipse and the mass sits in a focus [Kozlov, 1996] or whenever \(\Omega\) is made up of pieces of confocal quadrics with, again, the mass in a focus [Zhao,2024]. We show that the only analytic and centrally symmetric compact convex domains for which the corresponding Kepler billiards is analytically integrable are centred circles and focused ellipses, at least for sufficiently large energies. Joint work with Vivina Barutello, Irene De Blasi and Susanna Terracini.

On considère le problème de \(n\) corps dans le plan en interaction newtonienne avec des masses positives fixées, et l'on fixe le centre de masse à l'origine. Le système admet plusieurs intégrales premières, le moment cinétique \(C\), et l'énergie \(H\). On considère alors le système restreint à un niveau d'energie et de moment cinétique fixé. On présentera une notion d'intégrabilité pour les systèmes hamiltoniens restreints, et on montrera que le système n'est pas intégrable en ce sens, sauf peut-être pour \(H=C=0\).

In the \(n\)-body problem, the forces acting between particles approach infinity when the mutual distances approach zero. Therefore, at collision, the equations of motion have singularities. Since Levi-Civita and Sundman,the double collision has been ”regularized”, i.e. the singularity has been made to disappear by means of algebraic transformations. Levi-Civita obtained first a regularizing transformation of the Kepler problem based on the map \(z \mapsto z^2\) of the complex plane. This conformal map sends the orbits of the harmonic oscillator in the ones of the Kepler problem. The coordinates transformation is coupled with a slowing down of the motion by means of a time change defined by \(d\tau = 1/|z| dt\).

Based on classical results, the purpose of the work is the application of the regularization theory to optimal control. We consider the control of a spacecraft under the attraction of one or more bodies, where the control is the thrust, and with the goal of minimizing the integral of some function \(f\). If \(f = 1\), the setting is a time-minimization problem. The Pontryagin maximum principle (PMP) allows us to reduce the differential system to the analysis of an Hamiltonian system for finding optimal solutions. However, the system exhibits singularities in zero, and thus the PMP fails in the study of collision orbits. By applying the Levi-Civita regularization to the controlled system, we obtain an affine system (in the new time \(\tau\)) where the vector fields are complete and \(C^\infty\).

The modern asteroid surveys are producing very large databases of optical observations. Linking very short arcs (VSAs) of such observations we can compute the asteroid orbits. Searching for efficient linkage algorithms is an interesting mathematical problem.

We show how this problem can be faced using the first integrals of Kepler's problem.

Using all these integrals we find an overdetermined polynomial system of 9 equations in 6 variables, that is generically inconsistent, i.e. it does not have solutions, not even in the complex field, also when the two VSAs belong to the same observed body. We show how we can select two polynomial subsystems \(\mathcal{S}_1, \mathcal{S}_2\) that are still overdetermined, but consistent, and define two algebraic varieties with the lowest number of points (9 and 18 respectively). Moreover, we search for compromise solutions of both \(\mathcal{S}_1\) and \(\mathcal{S}_2\) using the concept of approximated gcd.

This is a joint work with Clara Grassi.

Joint work with Marcel Guàrdia and Tere M. Seara

The planar circular restricted three body problem (PCRTBP) models the motion of a massless body under the attraction of other two bodies, the primaries, which describe circular orbits around their common center of mass. In a suitable system of coordinates, this is a two degrees of freedom Hamiltonian system. The orbits of this system are either defined for all (future or past) time or eventually go to collision with one of the primaries. For orbits defined for all time, Chazy provided a classification of all possible asymptotic behaviors, usually called final motions.

By considering a sufficiently small mass ratio between the primaries, we analyze the interplay between collision orbits and various final motions and construct several types of dynamics.

We show that orbits corresponding to any combination of past and future final motions can be created to pass arbitrarily close to either one of the primaries. In particular, we also establish oscillatory motions accumulating to collisions. That is, oscillatory motions in both position and velocity, meaning that as time tends to infinity, the superior limit of the position and velocity is infinity while the inferior limit of the distance to one of the primaries is zero. Additionally, we construct arbitrarily large ejection-collision orbits (orbits which experience collision in both past and future times) and periodic orbits that are arbitrarily large and get arbitrarily close to either one of the primaries. Combining these results, we construct ejection-collision orbits connecting both primaries.

Résumé en français — Je commencerai par rappeler la méthode de résolution du problème de Kepler qui me semble la plus simple, due à Hamilton. J’indiquerai ensuite les solutions explicites de ce problème dans le cas des orbites rectilignes, commençant par une éjection du point mobile par le centre attractif et se terminant, soit à l’infini si E ≥ 0, soit par une collision du point mobile avec le centre attractif si E < 0, où E est l’énergie du mouvement. Je rappellerai la méthode de régularisation due à Moser, basée sur une idée de Fock (1935). Enfin je parlerai du difféomorphisme découvert par Ligon et Schaaf en 1976 (ou plutôt redécouvert, car il semble avoir déjà été mentionné par Györgyi en 1968) et j’expliquerai pourquoi ce difféomorphisme est symplectique.

Abstract in English — I will begin by recalling briefly the solution of the Kepler pro- blem due to Hamilton, which seems to me the easiest. Then I will indicate the explicit solutions of this problem for motions with a zero angular momentum, whose orbits are straight lines, which begin by an ejection of the moving point by the attractive center, and end, either at infinity if E ≥ 0, or by a collision of the moving point with the attrac- tive center if E < 0, where E is the energy of the motion. I will recall the regularisation method due to Moser, which rests on an idea of Fock (1935). Finlly I will discuss the diffeomorphism discovered by Ligon and Schaaf in 1976 (or rather re-discovered, since it seems to appear in a paper published by Györgyi in 1968) and I will explain why this diffeomorphism i symplectic.

All major planets in the Solar System (with the exception of Mercury) have almost circular orbits. This fact is not predicted by Kepler's laws and traditionally it has been suggested that, together with gravity, some small dissipative force should act.

In this talk we will consider a Kepler problem with friction
\[\ddot{x}+D(x,\dot{x}) \dot{x} =-\frac{x}{|x|^3},\; \; \; x\in \mathbb{R}^2 \setminus \{ 0\} \]
and we will find conditions on \(D\) in order to guarantee the circularization property. This means that the eccentricity of the orbits goes to zero as time evolves.

After some cylindrical and spherical blow up transformations, we obtain a system with an equilibrium. The circularization property is characterized in terms of the asymptotic stability of this equilibrium.

This is joint work with K. Kristiansen.

In this talk we deal, for the classical N-body problem, with the existence of action minimizing half entire expansive solutions with prescribed limit shape and initial configuration. Tackling the cases of hyperbolic, parabolic and hyperbolic-parabolic arcs in a unified manner, our approach is based on the minimization of a renormalized Lagrangian action defined on a suitable functional space. With this new strategy, we obtain a new proof for the already-known results of existence of both hyperbolic and parabolic solutions, and we prove for the first time the existence of hyperbolic-parabolic solutions for any prescribed asymptotic expansion in a suitable class. In addition, associated with each element of this class, we find a viscosity solution of the Hamilton-Jacobi equation as a linear correction of the value function. This work is in collaboration with Susanna Terracini.

In 1938, Williams [1] gave a group of interesting identities and inequalities of planar 5-body central configuration. But there are mistakes in the proof of inequalities. These inequalities for the strictly convex case were proved correctly by Chen and Hsiao [2] in 2017. Recently, with Professor Alain Albouy, we simplified the form of identities and proved the inequalities in the general case. This gives that the shifted Wintner-Conley matrix of planar 5-body central configuration has two positive non-zero eigenvalues.

[1] W. L. Williams, Permanent configurations in the problem of five bodies, Trans. Amer. Math. Soc. 44 (1938), 563–579

[2] K-C.Chen, J-S.Hsiao, Strictly convex central configurations of the planar five-body problem. Trans. Amer. Math. Soc. 370 (2018), 1907-1924

If a real-valued function \(f=f(x)\) has a nondegenerate critical point \(x_0\) then the implicit function theorem ensures that this critical point will survive under small perturbations of \(f\). On the other hand, if we assume that \(f\) has a manifold \(M\) of critical points then all these critical points are necessarily degenerate. However, under some nondegeneracy conditions on \(M\) as a critical manifold some of these points survive under small perturbations of \(f\). This well-known result can be generalized to the case where the perturbation is small only in the \(C^1\) sense and the ambient space is infinite-dimensional. In particular it can be applied to the search of periodic solutions of Hamiltonian systems, a problem already considered by Weinstein, Moser, and others in their generalizations of the Liapunov center theorem. It can also be used to recover some recent applications to the search of periodic solutions of the forced Kepler problem.

This is a joint work with R. Ortega.

This talk addresses the finiteness of central configurations in the planar N-body problem with \(-\alpha\)-homogeneous potentials. For the four-body problem, we establish that central configurations are finite in number for generic choices of masses when the exponent \(\alpha\) is rational. Furthermore, for positive masses and integer exponents \(\alpha>0\), finiteness holds whenever \(\alpha \neq 3k+1\) for any integer \(k \geq 0\). The finiteness of non-degenerate central configurations is established for all real \(\alpha > 0\), and a uniform upper bound independent of \(\alpha\) is derived.

Our approach combines fewnomial theory, the singular sequences technique introduced by Albouy and Kaloshin, and fundamental results from algebraic number theory.

Maxwell's ring model, which is a central configuration of the \((N+1)\)-body problom with \(N\) small satellites, was used to show the stability of the rings of Saturn by Maxwell(1859). Meyer & Schmidt(1993), Moeckel(1994), and so on, studied the related problem successively. In this talk, we consider the dominant mass is oblate. There are two kinds of periodic vertical libration, which are non-alternating and alternating cases. The Hamiltonian is highly symmetric and can be reduced to a simple form with two degrees of freedom. Then the Hamiltonian is expanded near the planar circular relative equilibria. By the Lie transformation technique, the Birkhoff norm form up to the fourth order is calculated. The vertical stability of such periodic orbits near the ring configuration is studied, according to Arnold's stability theorem. Some periodic orbits are also calculated.

• Chenciner Alain (Observatoire de Paris and Université Paris Cité)
  
• Niederman, Laurent (Université Paris Saclay & Observatoire de Paris PSL)
  

• Fejoz, Jacques (Université Paris Dauphine PSL & Observatoire de Paris PSL)
  
• Zhao, Lei (Universität Augsburg)
  

• Gruber, Jennifer (Institut für Mathematik, Universität Augsburg)
  
• Mulewski, Amélie (Laboratoire Temps Espace, Observatoire de Paris PSL)
  
• Pailler, Marine (Laboratoire Temps Espace, Observatoire de Paris PSL)