## Seminar (Differential equations)

Wintersemester 2012/13

Time and Place

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SE 2 std. | Mi 13:15-14:45 | 2A180 | 3.10 |

Topics

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.

Presentations

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:

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

Course assessment

Preparation and presentation of a chosen topic.

Audience

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

Auf Ihr Kommen freuen sich Aleksey Kostenko und Gerald Teschl