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

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)

(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

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