2021/2022

2021/2022

DM270

MAT/07 (MATHEMATICAL PHYSICS)

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

1°

2nd semester (01/03/2022 - 10/06/2022)

6

48 lesson hours

English in case of attendance of foreign students

ORAL TEST

MARZUOLI ANNALISA (titolare) - 6 ECTS

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

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.

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. Advanced topics (alternatively):
i) Canonical perturbation theory and overview of KAM (Kolmogorov, Arnold, Moser) theorem. ii) Poisson manifolds, bihamiltonian systems, Lax method and Toda system

Lectures

A. Fasano, S. Marmi “Analytical Mechanics: An Introduction”, Oxford University Press 2006
Notes on Poisson manifolds & Toda systems

Oral examination aimed to verify the assimilation of the basic notions and their interconnections.
Alternatively: two questions; presentation of a topic not addressed in the lectures

=