For many reasons, central configurations play a fundamental role in the study of the dynamics of the N-body problem. First, they are the only configurations through which some orbits pass that we can explicitly describe. Second, they arise naturally as limit shapes of total collision or completely parabolic expansion motions. They can be characterized as critical points of the Newtonian potential, and in this sense their determination gives rise to complex systems of polynomial equations, which appear treated from the beginning by Euler (for the case of three masses in the line [Eu]), until in the recent resolution in 2012 by Albouy and Kaloshin [AK] of the finitude problem (Smale problem for the 21st century [Sm]) for the case of 5 bodies in the plane. They also can be defined for all systems governed by homogeneous interaction potentials.
The main objective of this course is to introduce the student, through the construction of homographic orbits (and fitting in a certain way the Keplerian orbits within a system with three or more bodies), to the handling of the equations of motion, understanding the complexity of the dynamics, which is fundamental for the good assimilation of several of the advanced courses offered in this school. On the other hand, this introductory and preparatory course allows by itself to invite the student to a vast topic of current research.
The main reference for this course will be the well known notes by Rick Moeckel [Mo].
[Eu] L. Euler, De motu rectilineo trium corporum se mutuo attrahentium. Novi commen tarii academiae scientiarum Petropolitanae 11, 144–151 (1765). (read at St Petersburg in december 1763. Also in Opera Omnia S. 2, vol. 25, 281–289)
[AK] A. Albouy and V. Kaloshin, Finiteness of central configurations of five bodies in the plane. Ann. Math. (2) 176, 535–588 (2012)
[Sm] S. Smale, Mathematical problems for the next century. Math. Intell. 20 (1998),
Hamilton-Jacobi equation of N–body problems
Universidad de la República, Uruguay
The extension of the Aubry-Mather theory for convex Lagrangians to non-compact man ifolds, and then further allowing the Lagrangian to have singularities, naturally led to the notion of critical energy level in these systems, and in particular to the verification of the existence of weak global solutions of the critical Hamilton-Jacobi equation. These are the socalled weak KAM solutions. From this approach applied to the classical N-body problem [Ma], the abundance of completely parabolic motions emerged first. The critical level is the hinge between, on the one hand, the dynamics with chances of recurrence, and on the other, those that present expansion or scattering.
More recently, in work with A. Venturelli [MaVe], we have distilled the basis for the application of the Hamilton-Jacobi method in almost all Newtonian-type gravitational problems, using the now well known notion of viscosity solution. It is obtained, for example, that the Busemann functions associated with a given limit shape are of this type of solutions, and from them the existence of completely hyperbolic motions is deduced for positive energy levels, with arbitrary initial positions and arbitrary limit shapes.
In this course I will develop this approach from a geometric point of view. More precisely, the keys to this technique will be seen in terms of geodesic rays of the classic Jacobi Maupertuis metric, on which we are actively working today (see [PeSa][BuMa]).
[Ma] E. Maderna. On weak KAM theory for N–body problems, Ergod. Th. & Dynam. Sys. 32 (2012), 1019–1041.
[MaVe] E. Maderna, A. Venturelli. Viscosity solutions and hyperbolic motions: a new PDE method for the N–body problem, Annals of Mathematics 192 n.2 (2020), 499–550. [BuMa] J. M. Burgos, E. Maderna. On the geodesic rays of the positive energy levels in the Newtonian N–body problem, in preparation.
[PeSa] B. Percino, H. Sánchez-Morgado. Busemann functions for the N–body problem, Arch. Rational Mech. Anal. 213 (2014), 981–991.
KAM theory for a special kind of dissipative systems
Universidad Autónoma de México, México
The flows of mechanical systems with a friction added do not preserve the symplectic structure, but instead they contract the symplectic structure with a factor that does not depend on the phase variables. Geometrically these kinds of systems are conformally symplectic. In this advanced course, I plan to explain the developments in the KAM theory of conformally symplectic dynamical systems, hoping to cover KAM theorems for the existence of Lagrangian and lower-dimensional tori. In this dissipative setting, the KAM tori that one obtains are quasi-periodic attractors for the dynamics given fixed values of the parameters. I will explain the constructive theorems that several mathematicians have developed over the years [CCdlL13, CH14, HCF+16, CCdlL17, CCdlL20b, CCdlL20c], that prove the existence of quasi-periodic attractors in the dissipative setting of conformally symplectic systems. I will also present some of the numerical implementations that can be derived from the constructive theorems. See for example [CC10, CF12, CH17, BC19], for implementations for computing the attractors, their breakdown mechanisms and tori in the limit of small dissipation.
[CCdlL13] Renato C. Calleja, Alessandra Celletti, and Rafael de la Llave. A KAM theory for conformally symplectic systems: efficient algorithms and their validation, J. Differential Equations, 255(5):978-1049, 2013.
[CH14] Marta Canadell and Alex Haro. Parameterization method for computing quasi periodic reducible normally hyperbolic invariant tori, Advances in differential equations and applications, volume 4 of SEMA SIMAI Springer Ser., pages 85–94. Springer, Cham, 2014.
[CCdlL17] Renato C. Calleja, Alessandra Celletti, and Rafael de la Llave. Domains of analyticity and Lindstedt expansions of KAM tori in some dissipative perturbations of Hamiltonian systems, Nonlinearity, 30(8):3151–3202, 2017.
[CCdlL20b] Renato C. Calleja, Alessandra Celletti, and Rafael de la Llave. Existence of whiskered KAM tori of conformally symplectic systems, Nonlinearity, 33(1):538-597, 2020. [CC10] Renato Calleja and Alessandra Celletti. Breakdown of invariant attractors for the dissipative standard map, Chaos, 20(1):013121, 9, 2010.
[CF12] Renato Calleja and Jordi-Lluis Figueras. Collision of invariant bundles of quasi periodic attractors in the dissipative standard map, Chaos, 22(3):033114, 10, 2012. [CCdlL20a] R. Calleja, A. Celletti, and R. de la Llave. KAM estimates for the dissipative standard map, preprint (2020), available at arXiv 2002.10647.
[CCdlL20c] R. Calleja, A. Celletti, R. de la Llave, KAM theory for some dissipative systems, preprint: arXiv 2007.08394
Collision orbits for anisotropic singular problems
GIAN MARCO CANNEORI
Universit`a degli studi di Torino, Italy
Mechanical systems driven by anisotropic potentials have gradually become an object of interest in the theory of dynamical systems, initially for their applications in atom theory of crystal semicon- ductors, but also because they arise after particular reductions by symmetries of the classical N-body problem. The archetype model is the anisotropic Kepler problem, introduced by Gutzwiller in the 70s and further studied by Devaney in a series of remarkable papers (see [G, D1, D2, D3]). The peculiarity of such potentials resides in the presence of a non-constant angular component, which results in a complete loss of rotational invariance. As a consequence, the orbits structure is completely different from the case of the classical Kepler problem, the integrability is destroyed by the anisotropy and regularization methods for collision orbits inevitably fail.
In this course we will deepen the study of collision orbits for the anisotropic Kepler problem based on the Gutzwiller definition (see [G]). In particular, we will introduce McGehee coordinates to extend the flow beyond the singularity and we will analyse the dynamical features of the new flow (see [D1]). An exhaustive description of collision orbits will then be given, in order to show that they cannot be regularized. In fact, we will see how a particular method of regularization introduced by Easton in [E] cannot be employed in this situation and actually prove that the anisotropic Kepler problem is not regularizable (see [D2]).
For the purposes of this course we will mainly follow the notes of Devaney [D3].
[D1] R.L. Devaney. Collision orbits in the anisotropic Kepler problem. Invent. Math., 45(3):221–251, 1978.
[D2] R.L. Devaney. Nonregularizability of the anisotropic Kepler problem. J. Differential Equations, 29(2):252–268, 1978.
[D3] R.L. Devaney. Singularities in classical mechanical systems. In Ergodic theory and dynamical systems, I (College Park, Md., 1979–80), volume 10 of Progr. Math., pages 211–333. Birkhäuser, Boston, Mass., 1981.
[E] R. Easton. Regularization of vector fields by surgery. J. Differential Equations, 10:92–99, 1971.
[G] M.C. Gutzwiller. The anisotropic Kepler problem in two dimensions. J. Mathematical Phys., 14:139–152, 1973.
Chaotic dynamics in Celestial Mechanics
Universitat Polit`ecnica de Catalunya, Spain
The Restricted Three Body Problem is the simplest model in Celestial Mechanics: it models the motion of a body with negligible mass under the influence of two massive bodies which move on ellipses. Its dynamics is extremely rich and far from completely understood.
In this course I will explain how to construct chaotic dynamics for this model in different settings. To this end, we will introduce the classical Invariant manifolds and Melnikovs Theories, and we will show how to construct a Smale horseshoe (symbolic dynamics). We will apply all these techniques to the Restricted Three Body Problem. If time permitted, we will discuss how to apply these techniques when the model possesses different time scales and one has to face exponentially small phenomena.
Universidad Nacional Autónoma de México, México.
Universidad de la República, Uruguay.
Universitat Politècnica de Catalunya, Spain.
Universit`a degli Studi di Torino, Italy.
Juan Manuel Burgos
CINVESTAV (Ciudad de Mexico), Mexico.
Eddaly Guerra Velasco
Universidad Autónoma de Chiapas, México.
Laura Xiomara Gutiérrez Guerrero
MCTP (Chiapas), México.
Boris Percino Figueroa
Universidad Autónoma de Chiapas, México.
Renato Iturriaga Centro de Investigación en Matemáticas, México Email address: [email protected]