Teaching

Type:	Time:	Place:	Start:
SE 2 std.	Mi 13:15-14:45	2A180	3.10

The motion of a classical mechanical system is usually described in terms of Hamilton's equations. However, for most systems it is not possible to find the solution in closed form. However, if the system has a sufficient number of conserved quantities, one can construct a change of coordinates such that in the new variables (the action-angle variables) the Hamiltonian equations get linearized (the Liouville-Arnold theorem). Furthermore, many systems of interest are close to such an integrable system and one can show that the behavior of the integrable system persists for most initial conditions as long as the perturbed system is sufficiently close to the integrable one (the Kolmogorov-Arnold-Moser theorem).

Recently it was shown that these ideas, originally developed for finite-dimensional Hamilton's equations, also apply to certain partial differential equations which can be viewed as infinite-dimensional Hamiltonian systems. This has played an important role in the development of soliton theory and the prototypical integrable wave equation is the famous Korteweg-de Vries equation.

We will try to shed some light on this circle of ideas following the book of Kappeler and Pöschl. A solid background in analysis and differential equations will be assumed, but no previous knowledge about Hamiltonian systems will be neccesary.

Date:	Title:	Speaker:	References:
24.10	Overview	Aleksey Kostenko	[KP]
07.11	Background from differential geometry	Jonathan Eckhardt	[MR]
14.11	Hamiltonian Formalism	Katharina Kienecker	[KP]
21.11	Liouville Integrable Systems	Manfred Buchacher	[KP]
28.11	Birkhoff Integrable Systems	Alexander Beigl	[KP,M]
05.12	KAM Theory	Markus Holzleitner	[KP,W]
09.01	Background and Results	Aleksey Kostenko	[KP]

References:

1. T. Kappeler and J. Pöschl, KdV & KAM, Springer 2003.
2. J.E. Marsden and T.S. Ratiu, Introduction to Mechanics and Symmetry, 2nd ed., Springer 1999.
3. J. Moser, Lectures on Hamiltonian systems, Memoires of the Amer. Math. Soc. 81 (1968), 1-60.
4. C. E. Wayne, An Introduction to KAM Theory, Notes.

Preparation and presentation of a chosen topic.

Majors in Mathematics (master program, code MANS), Physics, ...