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.

Differential Geometry
(Ch. 1, §§ 1,2,3,4,5,7,8; A.1 e A.4)
Geometrical foundation of classical mechanics (Notes)
Hamiltonian flux, Liouville and Poincaré theorems (Cap. 8, §§ 3,5).
Symplectic structure of the phase space, Lie algebra of Hamiltonian matrices, symplectic group Hamiltonian vector fields
(Ch. 10, § 1)
Canonical transformation and their characerization; Poincaré-Cartan 1-form; generating functions
Ch. 10, §§ 2; 3, 4)
Algebraic structure of dynamical variables; Poisson brackets; Lie derivative, fluxes. Hamiltonian Noether theorem
(Ch. 10, §§ 5; 6; 9;)
Hamilton-Jacobi equations and examples; action-angle variables separability; Liouville theorem and Arnol'd hypotheses.
(Ch. 11, §§ 1; 2; 3; 4; 5; 6)
(*) Introduction to Poisson manifolds and the Orbit Method: see M. Audin "Spinning Tops"
Alternatively to (*):
Introduction to the canonical perturbation theory
(Ch. 12, §§ 1, 4, 5, 6)

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