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Physik | Wintersemester 2006/2007 |

06056 | Topics on integrable systems |

Dozent | Wendland K., Mencattini I. |

Dauer | 2 SWS |

Studiensemester | 7 |

Schein | Ja (Vortrag) |

Termin | Mo, 10:00-11:30, 1007/L1 |

Inhalt | The goal of these lectures is to give an introduction to the
theory of classical integrable systems. Even if integrable systems are highly non-generic dynamical systems, they seem to pervade the landscape of modern mathematics. In fact, the notion of integrability plays a fundamental role in the modern understanding of Huyghen's principle for hyperbolic PDE, in the theory of special functions (e.g. orthogonal polynomials and their generalizations), in algebraic geometry (e.g. the Kontsevich-Okounkov-Padharipande proof of Witten's conjecture about intersection theory on the moduli space of algebraic curves) and, finally, in number theory (e.g. the geometric approach to Langland's program). It is easier to give examples of integrable systems than to give a precise definition of integrability. Nevertheless, we can say that an integrable system is a hamiltonian system having the maximum possible number of (global) first integrals. For dynamical systems having a finite number of degrees of freedom, the property of being integrable has a nice geometrical interpretation: the phase-space is foliated by affine manifolds, where the motion of the system takes place. In some cases such affine manifolds are tori, where the hamiltonian flow linearizes. When we move to the case of infinite numbers of degrees of freedom, we lose the geometric intuition. In this case we can only say that the equations are to some degree soluble, that such solutions can be found explicitly and that there exist general methods to find them. Because the subject uses the methods of several different areas of mathematics (e.g. algebraic geometry, representation theory of Lie algebras and Lie groups) instead to dive in the deep of the theoretical background, I will focus the lectures on examples and I will discuss in detail the integrability of some remarkable dynamical systems. The lectures are meant to be elementary. A general knowledge of the rudiments of differential and algebraic geometry as well as of Lie groups and Lie algebras is welcome but not necessary. A tentative plan of the lectures is the following: Time permitting, the following topics could be discussed: |

Begleitend | 06057 |

Literatur | 1) Arnold, V.I. Mathematical methods of
classical mechanics. Graduate
Texts in Mathematics,60. Springer, Berlin (1984) 2) Audin, M. Spinning Tops. Cambridge studies in advanced mathematics, 51. Cambridge University Press (1996); 3) Reyman, A.G and Semenov-Tian-Shanski Group theoretical methods in the theory of finite dimensional integrable systems in Dynamical Systems VII, Encyclopedia of Mathematical Sciences, 16. Springer, (1994) 4) Guest, M. A. Harmonic maps, Loop Groups, and Integrable Systems. Student Text, 38. London Mathematical Society, (1997) 5) Ablowitz, M.J and Segur, H. Solitons and the Inverse Scattering Transform. SIAM, Studies in Applied Mathematics (1981) 6) Babelon, O. and Bernard, D. and Talon, M. Introduction to Classical Integrable Systems. Cambridge Monographs on Mathematical Physics, Cambridge University Press (2003). |

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- 14.03.2016