6 ECTS credits
150 h study time
Offer 1 with catalog number 4021564DNR for all students in the 1st semester at a (D) Master - preliminary level.
1) Definition of Hamiltonian systems, fundamental properties, examples.
2) Integrability in finite dimensions (Frobenius, Liouville, extensions), fundamental properties, examples and differences.
3) Important integrable Hamiltonian systemens in mathematics and physisc: sferical pendulum, rigid body, spinning top (Lagrange, Euler, Kovalevskaya), coupled spin-oscillators, coupled angular momenta...
4) Local behavior in regular points: theorem of Arnold-Liouville, transformation to action-angle coördinates.
5) Local beahviour in singular points: Eliasson-Miranda-Zung normal form for nondegenerate hyperbolic, elliptic and focus-focus components of singular points.
6) Semitoric systems: properties and interactions with the topology and geometry of the underlying manifold.
7) Integrability in infinite dimensions (“integrable Hamiltonian PDE”): motivation important examples (Korteweg-de Vries equation, Sine-Gordon equation, Nonlinear Schrödinger equation).
Essential information (schedule, literature/ lecture notes, homeworks etc.) will be posted on the webpage of the professor and/or assistant.
Good knowledge of the standard results and examples of finite dimensional integrable systems plus insights into infinite dimensional integrability. For details, see the list of content.
The final grade is composed based on the following categories:
Other Exam determines 100% of the final mark.
Within the Other Exam category, the following assignments need to be completed:
1 oral exam at the end of the semester (= in exam period),
several homeworks during the semester, graded by the assistent,
2nd chance exam for the oral exam, but not for the homeworks.
Grade = 75% oral exam + 25% homework results.
This offer is part of the following study plans:
Master of Mathematics: Fundamental Mathematics (only offered in Dutch)
Master of Teaching in Science and Technology: wiskunde (120 ECTS, Etterbeek) (only offered in Dutch)