Course Code : | 2600WETIHS |

Study domain: | Mathematics |

Academic year: | 2019-2020 |

Semester: | 1st semester |

Contact hours: | 60 |

Credits: | 6 |

Study load (hours): | 168 |

Contract restrictions: | No contract restriction |

Language of instruction: | English |

Exam period: | exam in the 1st semester |

Lecturer(s) | Sonja Hohloch |

At the start of this course the student should have acquired the following competences:

an active knowledge of

general notion of the basic concepts of

specific prerequisites for this course

an active knowledge of

- English

a passive knowledge ofIf everybody in the audience speaks Dutch the subject may/will be taught in Dutch. The lecture notes and/or literature are/stay nevertheless in English.

- English

- general knowledge of the use of a PC and the Internet

general notion of the basic concepts of

Information will be posted on the webpages of the professor and maybe partially in blackboard, so the student should be able to access this information.

specific prerequisites for this course

Necessary "pre-knowledge":

- Standard theorems of ordinary differential equations (ODE) and basic notions of dynamical systems (flow). One possible source are the (Dutch) lecture notes on "ODEs and dynamical systems" by Sonja Hohloch (---> webpage) or any standard text book on ODEs and dynamical systems.
- Standard notions of differential geometry (derivative, differential, tangent space...). Some elemantary explanations can be found in the (Dutch) lecture notes on "Multivariate Calculus" by Sonja Hohloch (---> webpage). Another possible source are the (Dutch) lecture notes on differential geometry by Tom Mestdag (---> webpage) or any standard text book on differential geometry.

- Knowledge and experience with finite dimensional integrable Hamiltionian systems (examples, standard properties).
- Knowledge and experience with local normal forms (regular/singular points) of integrable Hamiltionian systems.
- Knowledge and experience with toric and semitoric systems (common and distinguishing properties, classifications, topological/geometric interpretations).
- Knowledge and experience with infinite dimensional integrable systems (Korteweg-de Vries, Sine-Gordon, Nonlinear SchrÃ¶dinger equation).

- Definition of Hamiltonian systems, basic properties & examples.
- Integrability in finite dimensions (Frobenius, Liouville, more general notions), basic properties, examples.
- Important examples of integrable systems in mathematics and physics: Spherical pendulum, rigid body, top (Lagrange, Euler, Kovalevskaya), coupled spin oscillators, coupled angular momenta...
- Local behaviour (regular points): Theorem of Arnold-Liouville, transformation to action-angle coordinates.
- Local behaviour (singular points): local normal form of Eliasson-Miranda-Zung for nondegenerate singular points in terms of hyperbolic, elliptic, and focus-focus components.
- Toric systems: Symplectic classification by Delzant by means of polygons.
- Semitoric systems: properties and interaction with the topology and geometry of the underlying manifold.
- Integrability in infinite dimensions ("integrable Hamiltonian partial differential equations"): motivation and important examples (Korteweg-de Vries, Sine-Gordon, Nonlinear Schrödinger equation).

The course has an international dimension.

Class contact teachingLectures Practice sessions

Personal workExercises Assignments Individually

Personal work

Homeworks.

Examination

Continuous assessment

The professor will write lecture notes that cover the content of this class as presented during the lectures. These lecture notes are composed* during the term* and will be updated them more or less regular on the professor's WEBPAGE. [Since this class is taught for the first time, there do not yet exists specific lecture notes.]

There exists lots of literatur on integrable systems. The professor will post a selection of useful/helpful references ON HER WEBPAGE.

Concerning lectures: Sonja Hohloch, email sonja.hohloch AT uantwerpen.be

Concerning exercise sessions & homeworks: Marine Fontaine, email Marine.Fontaine AT uantwerpen.be