This course starts of from the case of finite dimensional Hamiltonian systems. We shall explain what integrability means in this context. We shall introduce the notion of Liouville integrability and state the Arnol'd-Liouville theorem which roughly speaking says that a system is integrable if admits ënough" (Poisson commuting) constants of motion. We shall then introduce another fundamental concept of modern mathematics: symmetries produce integrals of motion (Emmy Noether's theorem).
Before moving on to infinite dimensional system we shall study the example of the Manakov system trough its Lax pair. We shall show how from the Lax pair it is straightforward to obtain the needed constants of motion to prove integrability. Here too the role of the symmetries in the system is fundamental.
This example will lead us to consider the natural integrable systems which live on the coadjoint orbits of a Lie algebra. We shall then adapt this machinery to the case of pseudo-differential operators in order to study infinite dimensional systems such as the KdV equation. If there is enough time we'll study special solutions of the KdV such as solitons, finite gap solution and self-similar solutions.
Analytical mechanics: Hamiltonian and Lagrangian approach. Rigid body equations.
Lie algebras: main definitions of Lie algbera and Lie group. Adjoint and co-adjoint action.
Differential manifolds: tangent and cotangent bundle, vector fields, differential forms.
1) Finite dimensional Hamiltonian systems:
Recap on Poisson brackets and canonical transformations.
Notion of Liouville integrability.
Action angle variables for the pendulum.
Arnol'd Liouville theorem (no proof).
Example: solution of the Euler top by elliptic integrals.
2) Hamiltonian systems on coadjoint orbits:
Kostant-Kiriillov Poisson brackets.
Hamiltonian structure of
Example: Integrability of the Manakov system on so(n).
3) Infinite dimensional Integrable systems:
Lax pairs for KdV.