Students are advised to attend the MAGIC courses on Differentiable Manifolds 063 and on Lie Groups and Lie Algebras 008. Basic notions on Hamiltonian dynamics are needed.
Short introduction to finite dimensional Hamiltonian systems on Rn
. Poisson brackets.
Poisson manifolds, Casimir functions, Symplectic manifolds. Example: the symplectic structure on T∗
Generalised Darboux theorem. Hamiltonian vector fields and symplectic transformations.
Liouville integrability. Examples: n uncoupled harmonic oscillators. Toda chain (without proof of integrability). Arnol'd Liouville theorem. Action angle variables for the pendulum.
The role of symmetries in Integrability. Symplectic action of a Lie group on a Poisson manifold. Moment map. Noether Principle. Example: Kepler system and hydrogen atom.
Euler spinning top. Euler equation. Hamiltonian approach. Integration of the Euler top by elliptic integrals. Brief discussion on Liouville Arnol'd theorem for this specific case. Another approach: Lax equation for the spinning top.
Hamiltonian systems on coadjoint orbits:
Lie algebras. Lie Poisson brackets and Konstant Kirillow symplectic form. Lax pairs. Hamiltonian structure of Lax equations. Adapt the construction to loop algebras. Example: Integrability of the Manakov system on so(n).
Integrable models of interacting particles on the line:
Toda lattice and Calogero-Moser system.
Lax pairs and inverse scattering method.
Infinite dimensional Integrable systems:
The KdV equation. Pseudo-differential operators. Lax pairs for KdV. Infinite dimensional Poisson structure. Bi-hamiltonian structure of the KdV hierarchy. Symmetries. Solitons and scattering.