The orbits (unparametrized solutions) of a mechanical system q'' = f(q), where q is a point in an affine space, can be described in a 'projective' way viewing this affine space as homogeneous coordinates as has been developed in the works of Albouy. On the other hand, Kasner has given a number of curious geometric properties characterizing orbits of such mechanical systems, and in this talk we will examine some relations of these geometric properties to the structure of projective dynamics, as well as some analogous geometric properties for various types of projections of n-body systems.

In this talk, we present our work on the existence of noncollision singularities and superhyperbolic orbits and explain the roles that they play in understanding the global dynamics of the N-body problem.

Chazy showed that when a solution of the n-body problem tends to total collision then its normalized configuration converges to the set of normalized central configurations. In the planar problem, there are circles of rotationally equivalent central configurations. It's conceivable that by means of an "infinite spin", a total collision solution could converge to such a circle instead of to a particular point on it. Chazy proved that this is not possible if a certain nondegeneracy condition holds. I will discuss joint work with R. Montgomery, where we extended this to the degenerate case, at least if the limiting circle of central configurations is isolated from other circles of central configurations. (It is believed that all central configurations are isolated, even if they are degenerate, but this is not known in general.)  Our proof relies on combining  the center manifold theorem with the Łojasiewicz gradient inequality. The talk will also describe an explicit example of convergence to a degenerate central configuration of the planar four-body problem discovered by Palmore.

Les surprenantes découvertes d'anneaux autour de plusieurs petits corps du système solaire lointain ont ravivé l'étude de ces systèmes, jusqu'alors réservée aux planètes géantes. 

L'irrégularité des petits corps crée de fortes Résonances Spin-Orbite (RSO, dites aussi tessérales) entre le corps central et les anneaux. Ces résonances n'existent quasiment pas dans le cas des planètes géantes, dont le potentiel est essentiellement axisymétrique. Les petits corps sont donc des laboratoires naturels qui permettent de tester la réponse de disques à des résonances de forte intensité, et ce à des ordres élevés. 

Les RSO de premier ordre de type 1/2, 2/3,... (dites résonances de Lindblad) ont été largement étudiées dans les années 1960-80 dans le cadre de la dynamique galactique et des anneaux des planètes géantes. Ceci a permis la description mathématique des ondes spirales, et de leurs effets de confinement, voire de troncature avec bord net, du disque perturbé. 

Les résonances d'ordre supérieur (comme la RSO 1/3 d'ordre deux) ont été peu étudiées. D'une part parce qu‘elles imposent des développements non-linéaires complexes des équations hydrodynamiques, et d'autre part et surtout, parce que toute orbite résonante périodique d'ordre supérieur à deux se croise elle-même, conduisant à des singularités des équations hydrodynamiques. 

Dans ce contexte, il est intéressant de constater que les anneaux découverts autour de Chariklo, Hauméa et Quaoar sont tous proches de la résonance RSO 1/3. Nous présenterons des résultats de simulations N-corps qui explorent la réponse de disques collisionnels autour de cette résonance. Après une phase d'excitation qui conduit à l'auto-croisement des lignes de courant, l'anneau transfère l'énergie reçue de la RSO 1/3 vers des modes d’ordre un (1/2, 2/3, 3/4,...), ce qui conduit in fine à un fort confinement de l'anneau, via des mécanismes non linéaires qui restent à élucider.

In this presentation, I will share a series of recent findings developed through collaborations with L. Ruiz, G. Gevorgyan, G. Boué, A. Correia, and I. Matsuyama. Initially, I will introduce the concept of Love numbers within the frequency domain to frame our discussion on tidal interactions. Subsequently, I will outline the time-domain mathematical model we have refined over several years, demonstrating its capability to capture the tidal response of multilayered planetary bodies. Through our model, I will highlight a range of phenomena critical to understanding real-world tidal interactions, although not currently encompassed by our framework. Lastly, I will outline a possible  methodology for calibrating our model's parameters utilizing observational data, underscoring a potential key advantage of our approach.

We are concerned with the possibility of modeling a given dynamical system by patching different dynamics, simpler than the given one. One interesting case is when the simpler dynamics correspond to integrable systems. We shall discuss the results of a recent work (Cavallari, Gronchi, Baù, Physica D 2022) about the case of the Sun-shadow dynamics, i.e. the planar motion of a mass particle in a force field defined by patching Kepler's and Stark's dynamics: this can be seen as a basic model for the motion of an Earth satellite perturbed by the solar radiation pressure and considering the Earth shadow effect. 

The existence of periodic orbits of brake type is proved, and the Sun-shadow dynamics is investigated by means of a Poincaré-like map defined by a quantity that is not conserved along the flow. We also present the results of our numerical investigations on some properties of the map. Moreover, we construct the invariant manifolds of the hyperbolic fixed points related to the periodic orbits of brake type. The global picture of the map shows evidence of regular and chaotic behaviour.