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Lectures and Abstracts

Singularities and collisions of solutions to the N-body problem

by Susanna Terracini (Università di Torino)


Through the study of singularities and collisions , we provide an introductory course to the N-body problem and more generally to Hamiltonian dynamics.


Closed orbits for twisted autonomous Lagrangian systems

by Gabriele Benedetti (University of Münster)


In this series of talk we study the problem of existence of periodic orbits of fixed energy for twisted autonomous Lagrangian systems on closed manifolds. We begin with relating such orbits with the zeros of the free-period action form. We discuss its general properties with particular attention given to the compactness of critical sequences and to the completeness of a suitable flow transverse to the action form. Then, we combine such results with convenient minimax methods to prove old and new existence results for the zeros of the action form and we discuss the relevance of stable energy levels in this
context. Finally, we look in more detail to the special case of magnetic flows on closed surfaces: first, we present a method originally due to Taimanov to find periodic orbits with small energy which are local minimizers of the action form; second, we give some sufficient criteria in order to verify stability in this setting.

 

 

Topics in hyperbolic dynamics

 

Part 1 by Rafael Potrie (Universidad de la República)

Introduction to non-uniform and partial hyperbolicity

We shall present the theory of non-uniformly hyperbolic diffeomorphisms trying to concentrate in some simplified contexts and explain some of the main techniques in the field. Some of the topics include: Lyapunov exponents, Invariant manifolds (Pesin theory and persistence properties) and dynamical consequences. The topics will help introduce some concept for the second part of the minicourse but will also cover some topics of independent interest.


Part 2 by Marie-Claude Arnaud (Université d'Avignon)

Hyperbolicity for conservative twist maps

We will present Birkhoff's and Aubry-Mather theory for the conservative twist maps of the 2-dimensional annulus. We will then focus on what happens close to the Aubry-Mather sets: definition of the Green bundles, link between hyperbolicity and shape of the Aubry-Mather sets, behaviour close to the boundaries of the instability zones.  We will finish by giving some open questions.

 

 

Homoclinic orbits, non-integrability, chaos and global instability in Hamiltonian systems

by Marcel Guardia and Tere M. Seara (Universitat Politècnica de Catalunya)

 

TIn this course we present some results about instabilities in the restricted three body problem. The restricted three body problem can be seen as a perturbation of the two body problem in different settings. A feature of the three body problem is that it is integrable and that its orbits lie on conic sections. Moreover, the angular momentum is preserved along the orbits.
We will see that the restricted three body problem presents different unstable behaviors. First, it exhibits oscillatory motions, that is,  orbits  which leave every bounded region but which return infinitely often to some fixed bounded region. We will see that the restricted three body problem has also orbits with a large drift in the angular momentum. This last phenomenon is known as Arnold difusion. The methods to prove the existence of these two types of motions rely in the so called geometric methods. Both are consequence of the fact that the invariant manifolds of infinity intersect transversally when the angular momentum is big.

 

 

The principle of minimal action in dynamics and geometry

by Alfonso Sorrentino (Roma Tre)

 

In this series of lectures we shall present John Mather's variational approach to the study of convex and superlinear Hamiltonian systems, what is generally called Aubry-Mather theory. Starting from the crucial observation that invariant Lagrangian graphs can be characterised in terms of their "action-minimizing properties", we shall investigate how analogue features can be traced in a more general setting, namely the so-called Tonelli Hamiltonian systems. This different point of view will bring to light a plethora of compact invariant subsets of the system that, under many points of view, could be considered as  generalisation of invariant Lagrangian graphs, despite not being in general either submanifolds or regular. We shall discuss their structure and their symplectic properties, as well as their relation to the  dynamics of the system. Moreover, time permitting, we shall point out  some connections of this theory to other topics, such as classical mechanics, Hamilton-Jacobi equation (weak KAM theory), symplectic geometry, Hofer's geometry etc.

 

 

 

Generic Mañé sets

by Gonzalo Contreras (CIMAT Mexico)

 

We prove that for a Tonelli Lagrangian L on a closed surface M, there is an open and dense subset G of C2(M,R) such that for any f in G, the Lagrangian L(x,v) + f(x) has a unique minimizing measure and this measure is supported on a hyperbolic periodic orbit.

 

 

Generic dynamics close to homoclinic bifurcations, in high regularity

by Nicolaz Gourmelon (Université de Bordeaux)

 

We characterize the dynamics that appear close to homoclinic bifurcations smoothly generically, in particular close to homoclinic tangencies, depending on the Lyapunov exponents and of the indices of domination along the tangency. The dynamics we characterize are Newhouse phenomena, universal dynamics of some dimensions, heterodimensional cycles, blenders, etc..

 

 

Periodic orbits of exact magnetic flows on surfaces

by Marco Mazzucchelli (ENS Lyon)

 

This talk is about periodic obits of exact magnetic flows on the cotangent bundle of closed surfaces. The dynamics of these Hamiltonian systems on high energy levels is well known: it is conjugated to a Reeb flow, and actually to a Finsler geodesic flow. In this talk, I will focus on low energies, more precisely on energies below the so-called Mañé critical value of the universal covering. After introducing the setting, I will present a recent result asserting the existence of infinitely many periodic orbits on almost all energy levels in this range. This is a joint work with A. Abbondandolo, L. Macarini, and G. P. Paternain.

 

 

Critical Busemann functions for the N-body problem.

by Boris Percino (UNAM Oaxaca)

 

Following ideas of Maderna and Venturelli, we prove that the Busemann function of the parabolic homotetic motion for a minimal central coniguration of the N-body problem is a viscosity solution of the Hamilton-Jacobi equation and that its calibrating curves are asymptotic to the homotetic motion. Joint work with Héctor Sánchez-Morgado.

 

 

Existence of the Hénon family in the newtonian

three-body problem with two small equal masses.

by Andrea Venturelli (Université d'Avignon)

 

In 1975, M. Hénon found numerically a remarkable one-parameter family of reduced periodic
solutions, in the planar newtonian three-body problem with equal masses. Solutions of this family are periodic in a rotating frame. This family can be parametrized by angular momentum C. When C=0, one gets Schubart's collinear solution. Increasing C, for some value C* one gets a fully periodic solution, called Broucke-Hénon solution. We prove the existence of this family in the three-body problem, with one mass equal to one and two small equal positive masses. It is a classical problem of continuation of periodic solutions. It is a degenerate problem, but exploiting symmetries, we are able to recover nondegeneracy, and to prove the existence of this family using implicit function theorem. It is a work in progress with Anete Soares.

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