# 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)

*T*In 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.

*
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.*

*
*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..

*
**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.*

*
*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.

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.