AM550: Problemi di piccoli divisori


Dynamical systems and vector fields on finite and infinite dimensional Hilbert spaces. Vector fields as derivations, push-forward of a vector field, Lie derivative, the Lie exponentiation formula.  Symmetries and constants of motion. Reducibility of linear quasi-periodically forced systems. Finite and infinite dimensional case.  Hamiltonian vector fields on a finite and infinite dimensional Hilbert space.
Hamiltonian formalism, symplectic changes of variables. Analytic Hamiltonians,  regular Hamiltonians, norms and  Cauchy estimates. Polynomial hamiltonians and degree decomposition. 
KAM theory for finite dimensional tori: maximal tori, elliptic tori.

Suggested textbooks

 Moser- Zehnder:  Notes on dynamical systems.