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 foundation of Lagrangian and Hamiltonian mechanics. Hamiltonian flux, Liouville’s and Poincaré’s 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’s theorem). Hamilton-Jacobi equations; action-angle variables in the 1-dimensional case and in the n-dimensional, separable case. Completely integrable Hamiltonian systems: Liouville’s and Arnol’d’s theorems. Canonical perturbation theory and an overview of KAM (Kolmogorov, Arnol’d, Moser) theorem. Poisson manifolds and the method of coadjoint orbits.



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 Hamiltonianmatrices, 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