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Accueil > Écoles de recherche > Ecoles de recherche 2015 > Liste chronologique des écoles de recherche 2015 > Dynamical systems : Examples, billiards, and the 3-body problem

Dynamical systems : Examples, billiards, and the 3-body problem

English Version

CIMPA-NIMS-SOUTH KOREA research school

Daejeon, May 25-June 3, 2015

Report by Urs Frauenfelder

Report by Felix Schlenk

The goal of this school is to introduce the students
to basic questions and methods in dynamical systems through
concrete examples.
Our basic questions are : Existence of cloded orbits, integrability
(ie, complete solvability), and stability.
And the methods are : geometrization and variational principles.
We shall do this by illustrating these questions and methods
by elementary and archetypical examples of dynamical systems :
Billiards, geodesic and magnetic flows, and the 3-body problem.

Another goal is to show that these basic examples and the question we ask
are by no means dry mathematical abstractions, but have their origin in
concrete problems such as mechanics, geometrical optics, and celestial mechanics.

Administrative and scientific coordinators

- Urs Frauenfelder (Seoul National University) frauenf@snu.ac.kr
- Felix Schlenk (Neuchâtel) felix.schlenk@unine.ch

Scientific Committee

  • Seonhee Lim (Seoul National University)
  • Felix Schlenk (Neuchâtel)
  • Ana Rechtman (Strasbourg)
  • Urs Frauenfelder (Seoul National University and Münster)
  • Otto van Koert (Seoul National University)

Local Organizing Committee

  • Seonhee Lim (Seoul National University) slim@snu.ac.kr
  • Urs Frauenfelder (Seoul National University and Münster) frauenf@snu.ac.kr
  • Otto van Koert (Seoul National University) okoert@snu.ac.kr
  • Felix Schlenk (Neuchâtel) felix.schlenk@unine.ch

Scientific Program

There will be a 6-hours precourse by Otto van Koert and Felix Schlenk : Definition of geodesics, geodesics on round spheres and real projective space, the flat torus, surfaces of revolution (the Clairaut Integral), Kepler’s 2-body problem ; Definition of the billiard map, billiards in rectangles (unfolding, each orbit is either closed or uniformly filling), billiards in the circle (each orbit is either closed or with bounce points uniformly distributed on the boundary)

Then there will be five more specialized courses, by Youngjin Bae, Urs Frauenfelder, Otto van Koert, Seonhee Lim, Ana Rechtman and Felix Schlenk.

Each specialized course will have two 1-hour sessions of exercises.

All lectures are in English.

Abstracts

Precourse Otto van Koert and Felix Schlenk

Definition of geodesics, geodesics on round spheres and real projective space, the flat torus, surfaces of revolution (the Clairaut Integral), Kepler’s 2-body problem ; Definition of the billiard map, billiards in rectangles (unfolding, each orbit is either closed or uniformly filling), billiards in the circle (each orbit is either closed or with bounce points uniformly distributed on the boundary)

Examples Seonhee Lim

The geodesic flow, a non-round metric on S^2 all of whose geodesics are closed, magnetic flows, magnetic monopoles, Arnold’s example on T^4, closed geodesics on hyperbolic surfaces are dense in phase space, the horocycle flow has no closed orbits, the rotating 3-body problem

Billiards Youngjin Bae

The billiard map is measure preserving, billiards in ellipses, two applications (a light trap and a non-illuminatable room), The Poincaré recurrence theorem, non-existence for light-traps for non-parallel rays, Caustics, existence of caustics in smooth stricly convex billiards via KAM, Geodesics on the 3-ellipsoid, and relation to billiard orbits

Variational Methods Felix Schlenk

The mountain pass theorem in R^n, broken geodesics, Existence of a geodesic on S^2 for any Riemannian metric, existence of closed orbits in a convex billiard (for any number p of bounces, with rotation number q < p/2), relation to Poincaré’s Last Geometric theorem

Flows without periodic orbits and plugs Ana Rechtman

The problem of determining when the flow of a non-singular vector field on a closed 3-manifold has a periodic orbit has a long history. We will study examples of vector fields whose flow has no periodic orbits on any closed 3-manifold. These were first constructed by P. A. Schweitzer for C^1 vector fields, and then by K. Kuperberg in the smooth and real analytic categories. Schweitzer’s construction was then achieved in the volume preserving category by G. Kuperberg, giving C^1 volume preserving vector fields without periodic orbits. An open question is whether a volume preserving flow on a closed 3-manifold must have periodic orbits. The construction of these flows are all based on the use of plugs : a devise that allows to destroy periodic orbits. In the course I will present the constructions and main applications.

The restricted 3-body problem Urs Frauenfelder

The restricted 3-body problem in inertial coordinates. The restrichted 3-body problem in rotating coordinates and the Jacobi integral. Hill’s lunar theory. Regularization. Periodic orbits in the restricted three body problem : -The direct and retrograde period orbit. -The Lyapunov periodic orbit. -The antisymplectic involution and symmetric periodic orbits. -Birkhoff’s shooting method. Global surfaces of section -Existence of global surfaces of section according to Poincarée, Conley, and McGehee. -Area preserving disk and annulus maps. Results of Birkhoff and Franks-Handel. Homoclinic and Heteroclinic periodic orbits in the restricted three body problem and its applications to space mission design. Outlook on holomorphic curve techniques in the restricted three body problem and the work of Hofer, Wysocki, and Zehnder.

Deadline for registration

January 6, 2015

Application procedure only for applicants not from South Korea.

Applicants from South Korea should send an e-mail to the address okoert@snu.ac.kr with the following information : complete name, name of home institution, short letter of motivation.

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