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SYMPLECTIC MECHANICS - 6 CFU

Teacher

Francesco Fasso'

Scheduled Period

I Year - 2 Semester | 28/02/2022 - 11/06/2022

Hours: 48 (24 esercitazione, 24 lezione)

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Prerequisites

Basic notions of Differential Geometry (manifolds, differential forms, vector fields), at the level at which they are treated in the course "Differential Geometry" at the first semester. Some knowledge of Lagrangian and Hamiltonian mechanics (at the level of the course "Fisica Matematica" of the II year of the Laurea Triennale) is useful but not strictly necessary.

Target skills and knowledge

The course provides an introduction to symplectic geometry, Hamiltonian mechanics on symplectic manifolds, Lie groups and their actions. Special attention is given to symmetry and
integrability. The emphasis is on the differential and topological aspects of the subject.

Examination methods

Oral examination on the topics treated in the course.

Assessment criteria

Evaluation of the knowledge and of the mathematical comprehension of the subject.

Course contents

1. Symplectic manifolds and Hamiltonian systems.
Review of Hamiltonian mechanics in R^(2n). Symplectic forms and
symplectic geometry. Cotangent bundles. Hamiltonian vector fields and their Lie algebra. Symplectic maps and generating functions. Hamilton-Jacobi equation and Lagrangian submanifolds. Poisson manifolds. Examples.

2. Lie groups and their actions.
Lie groups and their geometric structure (Lie algebras, exponential map, maximal tori). The Lie-Poisson structure of the dual of a Lie algebra; coadjoint orbits. Lie group actions. Quotient spaces. Principal bundles. Examples.

3. Symmetry, reduction and conservation laws.
Invariant vector fields. Reduction. Reconstruction. Relative equilibria. Hamiltonian actions on symplectic manifolds. Lifted actions on cotangent bundles. Momentum maps. Symplectic and Poisson reduction. Left-invariant Hamiltonian systems on cotangent bundles of Lie groups. Examples.

4. Integrability.
Torus actions and integrability. The Lioville-Arnold theorem and some generalizations. Examples.

Planned learning activities and teaching methods

Lectures and tutorials

Additional notes about suggested reading

Selected parts from textbooks will be indicated, and written notes will be distributed, during the course.

Textbooks (and optional supplementary readings)

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