An introduction to the geometry of integrable Hamiltonian systems SSD MAT07 12 ore / 2 ECTS The minicourse will take place between the end of October and the beginning of November

An important question in Hamiltonian mechanics is to describe qualitative properties of the dynamics under consideration. In general, this is a hard problem but it can be tackled for those systems that are known to be integrable, i.e. that admit the largest number of constants of motion. In spite of their relatively simple dynamical behaviour, integrable Hamiltonian systems play a prominent role in Hamiltonian mechanics and beyond, ranging from symplectic geometry, Lie theory and quantum mechanics. The aim of this course is to provide an introduction to the geometry of such systems from a symplectic perspective. After introducing the necessary tools from symplectic geometry, we will study the structure of integrable Hamiltonian systems near regular points and regular (connected components of) fibres, proving the Darboux-Carathéodory and the Liouville-Arnol'd theorems. The theory will be illustrated by some (simple) examples.