Physik Wintersemester 2006/2007
06056 Topics on integrable systems [SV]
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:
1) Introduction to symplectic and Poisson geometry
2) Integrability {it $acute{a}$ la} Liouville, Arnold's theorem and action-angle variables
3) Synopsis of algebraic and analytical methods ($r$-matrices, AKS scheme, spectral curves)
4) Toda systems and spinning tops
5) Hamiltonian reduction, projections' method
6) Rational and trigonometric Calogero-Moser's systems
7) KdV's equation and its interpretation as hamiltonian system
8) Remarks on the inverse scattering method, action-angle variables, multi-solitons and rational solutions for KdV's equation
9) KdV's equation as an Euler's equation
10) Calogero-Moser's system and rational solutions of the KdV's equation

Time permitting, the following topics could be discussed:
10) Integrable hierarchies and Sato-Segal-Wilson's grassmannian
11) Bispectral problem.

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).