The JISD2004 will be devoted to the two courses

• Asymptotic Methods In Dynamical Systems (48114), by Prof. Luigi Chierchia, Università degli Studi ``Roma Tre", on the topic Quasi-periodic solutions for the three-body problem.
• Seminar of Hamiltonian Systems and Celestial mechanics (48036), by Prof. Alain Chenciner, Institut de Mécanique Céleste, Paris, on the topic Calculus of variations and introduction to weak KAM theory.

The JISD2004, as well as the Doctoral Programme in Applied Mathematics, is supported by a Spanish grant Mención de calidad en programas de doctorado .

Contents

• Luigi Chierchia, Quasi-periodic solutions for the three-body problem

  1. Hamiltonian formulation for nearly-integrable three-body problems.
  2. KAM theory.
  3. Maximal quasi-periodic solutions and total stability for the restricted, planar, circular three-body problem.
  4. An overview of extensions, perspectives and open problems.

• Alain Chenciner, Calculus of variations and introduction to weak KAM theory

  1. In a first part, the study of the variations of an action integral of the form leads to a natural introduction of the basic notions of Classical Mechanics: Legendre transform, Hamiltonian, Poincaré-Cartan integral invariant, symplectic and contact structures, Hamilton-Jacobi theory.
  2. In the second part, one restricts the attention to convex and superlinear Lagrangian, the ones for which exists a good theory of minimization, in particular Tonelli's theorem. In the ``weak KAM theory" (Fathi's terminology), one uses minimization to find global weak solutions of the Hamilton-Jacobi equation which arise in many domains of mathematics, for example, Aubry-Mather theory, viscosity solutions, etc.

Schedule
 

Monday June 28	09.00 - 11.00	Calculus of variations in the convex case (local structures). From Euler-Lagrange equations to the Poincaré-Cartan integral invariant, the Legendre transform and Hamilton's equations. Exercices: Flows, differential forms, symplectic structures. (Alain Chenciner)
	11.00 - 11.30	Cofee break
	11.30 - 13.30	Delaunay symplectic variables for the two body problem. Formulation of the planar, restricted three-body problem and analyticity properties of the Hamiltonian. Part I. (Luigi Chierchia)
Tuesday June 29	09.00 - 11.00	The Hamilton-Jacobi equation. The solutions of Hamilton's equations as characteristics. Lagrangian submanifolds and geometric solutions of the Hamilton-Jacobi equation. Caustics as an obstruction to the existence of global solutions to the Cauchy problem. Exercices: The geodesic flow on a torus of revolution as an example of a completely integrable system. (Alain Chenciner)
	11.00 - 11.30	Cofee break
	11.30 - 13.30	Delaunay symplectic variables for the two body problem. Formulation of the planar, restricted three-body problem and analyticity properties of the Hamiltonian. Part II. (Luigi Chierchia)
Wednesday June 30	08.00 - 10.00	Minimizers. Weierstrass theory of minimizers. Minimizing KAM tori. Existence of minimizers (Tonelli's theorem) and the Lax-Oleinik semi-group. Exercices: Examples around the pendulum. (Alain Chenciner)
	10.00 - 10.15	Cofee break
	10.15 - 12.15	A classical isoenergetic KAM theorem and application to the planar, circular, restricted three-body problem. Part I. (Luigi Chierchia)
Thursday July 1	09.00 - 11.00	Global solutions of the Hamilton-Jacobi equation. Weak KAM solutions as fixed points of the Lax-Oleinik semi-group; convergence of the semi-group in the autonomous case. Conjugate weak KAM solutions. Exercices: Burger's equation and viscosity solutions. (Alain Chenciner)
	11.00 - 11.30	Cofee break
	11.30 - 13.30	A classical isoenergetic KAM theorem and application to the planar, circular, restricted three-body problem. Part II. (Luigi Chierchia)
Friday July 2	09.00 - 11.00	Mather's theory. Class A geodesics and minimizing measures. The a and b functions as a kind of integrable skeleton. Exercices: Hedlund's examples. The time-periodic case as a generalization of Aubry-Mather theory. Exercices: classical Aubry-Mather in Birkhoff billiards. (Alain Chenciner)
	11.00 - 11.30	Cofee break
	11.30 - 13.30	Extensions (lower dimensional tori, many bodies,...). An overview. (Luigi Chierchia)

(*) All the courses will be held in the room n.005 of the FME building (Facultat de MAtematiques i Estadistica), at C/ Pau Gargallo, n. 5 Barcelona, 08028.  
 

For more information, please contact Tere.M-Searaupc.edu

April 04 -RMC