MAGIC067: Integrable Systems

The course is an introduction to the theory of integrable systems.

We will consider mainly the finite-dimensional Hamiltonian systems with integrability understood in Liouville's sense.

The content covers both classical techniques like separation of variables in the Hamilton-Jacobi equation as well as modern inverse spectral transform method.

The main examples include Kepler problem, geodesic flow on ellipsoids, Euler top, Toda lattice, Calogero-Moser system and Korteweg- de Vries equation. 

Students are advised to attend the MAGIC courses on Differentiable Manifolds 063 and on Lie Groups and Lie Algebras 008.

Hamiltonian systems and Poisson brackets. Integrals and symmetries, Noether principle. Example: Kepler system.  Integrability in Liouville's sense. Liouville-Arnold theorem, action-angle variables. Example: anisotropic harmonic oscillator.

Hamilton-Jacobi equation and separation of variables. Geodesics on ellipsoids and Jacobi inversion problem for hyperelliptic integrals.

Euler equations on Lie algebras and coadjoint orbits. Multidimensional Euler top, Manakov's generalisation and Lax representation.

Toda lattice and inverse spectral transform method. Direct and inverse spectral problems for Jacobi matrices and explicit solution to open Toda lattice.

Calogero-Moser system and Hamiltonian reduction. Scattering in Calogero-Moser system.

Korteweg-de Vries equation as an infinite-dimensional integrable system. Integrals and Hamiltonian structures, Lenard-Magri scheme.