Second Level Degree in Mathematics
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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)
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)
- J.E. Marsden and T.S. Ratiu, Introduction To Mechanics And Symmetry. 2nd edition., Springer, 2010.
- Arnolʹd, V. I., Mathematical methods of classical mechanics, Springer Verlag, 1989.
- F. Cardin, Elementary Symplectic Topology and Mechanics, Springer Verlag, 2015.
- McDuff, Dusa, Salamon, Dietmar, Introduction to symplectic topology, Oxford Mathematical Monographs, 1998.