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### The Theory of Dynamical Systems

Professors:
Marzuoli Annalisa
Year:
2015/2016
Course code:
500702
ECTS:
6
SSD:
MAT/07
DM:
270/04
Lessons:
48
Period:
I semester
Language:
Italian

#### Objectives

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

#### Prerequisites

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

#### Syllabus

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)

#### Bibliography

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

Dipartimento di Matematica ''F. Casorati''

Università degli Studi di Pavia - Via Ferrata, 5 - 27100 Pavia
Tel +39.0382.985600 - Fax +39.0382.985602