Resonances in Dynamical Systems

Reading Course

Course Details:
  • Lecture every Thursday, 13-15. Last lecture on Wednesday (!), April 05 15-17.
  • Until further notice, the lecture will take place in room 332B.
  • Course Notes (updated 17th March 2023)
Seminar:

The seminar part of this course will follow in may with the following talks.

KAM theory for infinite-dimensional Hamiltonian system
15/05/2023 15-17 Introduction to the Hamiltonian structure of nonlinear wave equations Periodic and Quasi-periodic Solutions of Nonlinear Wave Equations via KAM Theory (part I)
17/05/2023 15-17 Periodic and Quasi-periodic Solutions of Nonlinear Wave Equations via KAM Theory (parts II and III)
22/05/2023 15-17 KAM Tori for 1D Nonlinear Wave Equations with Periodic Boundary Conditions (parts I and II)
24/05/2023 15-17 Introduction to the KdV equation and its complete integrability
25/05/2023 13-15 A KAM-Theorem for Equations of the Korteweg-de Vries Type (parts I and II)
Abstract:

At least since the discoveries of Isaac Newton, mathematicians have wondered about the stability of celestial bodies attracted by gravitational forces. While the sun’s mass is roughly a thousand times bigger than the mass of the biggest planet Jupiter (which itself has a mass rougly \(10^2\) to \(10^3\) orders larger to the other planets), a perturbative approach to the \(n\)-body problem as a system of many solvable two-body problems fails in general. The reason for this are resonances between the planets when the ratio of orbital periods forms a small fraction. A perturbative approach to the \(n\)-body problem leads to the appearance of small denominators. For this reason, the study of convergence of the corresponding series has been an open problem until the major breakthrough known as KAM theory (after Kolmogorov, Arnold and Moser).

KAM theory provides stability of the corresponding invariant tori for certain perturbations if the resonances can be controlled. This course will be a gentle introduction into the problem of small divisors and its resolution. We study KAM theory first phenomenologically for simple dynamical systems such as perturbations of rotations of the circle or twist maps. Then, we continue with finite-dimensional integrable Hamiltonian systems.

More modern advances of KAM theory include the application to infinite-dimensional dynamics. We will discuss problems coming from the theory of water waves or quantum mechanics.

Course Structure:

The course is primarily intended for Ph.D. students and will proceed as a weekly lecture during reading periods 3 and 4. The basic tools of KAM theory will be developed in the first half of the course by myself, first based on examples and then in the context of (finite-dimensional) Hamiltonian systems. The second part of the course consists of talks given by the participants: there the transfer of KAM methods to the infinite-dimensional examples in applications such as the theory of water waves or quantum mechanics will be discussed. Lecture notes will be posted on my webpage as the term proceeds.

Preliminary Meeting:

There will be a preliminary meeting before Christmas to decide on the topics for the second half of the course, as well as the organisation. The first part of the course will be rather self-contained. I will provide introductions into the models discussed in the second part of the course during the lecture. Please fill in the Google Form until this Thursday, December 08!