Aim of the course is to make the students acquainted with advanced topics in Analytical Mechanics. A few subjects in the last part of the course will be chosen in agreement with the students'preferences.

Lectures

Oral Examination

A course of Analytical Mechanics (Lagrangian and Hamiltonian formulations). Basic knowledge of differential geometry would be helpful.

Geometrical foundations of Lagrangian and Hamiltonian mechanics. Hamiltonian flux, Liouville and Poincaré theorems. Symplectic structure on the Hamiltonian phase space; Poincaré-Cartan 1-form and symplectic form. Canonical transformations and their characterization. Algebraic structure of dynamical variables: Poisson brackets and relations with Lie derivatives. Constants of motion and symmetry properties (Hamiltonian Noether theorem). Hamilton-Jacobi equations; action-angle variables in the 1-dimensional case and in the n-dimensional, separable case. Completely integrable Hamiltonian systems: Liouville and Arnold theorems. Advanced topics:

i) Canonical perturbation theory and overview of KAM (Kolmogorov, Arnold, Moser) theorem. ii) Poisson manifolds, the method of coadjoint orbits and introduction to geometric quantization; iii) Algebraic-topological methods in the study of discrete dynamical systems.

A. Fasano, S. Marmi "Analytical Mechanics: An Introduction", Oxford University Press 2006