Second Level Degree in Mathematics
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SYMPLECTIC MECHANICS - 6 CFU
Teacher
Franco Cardin
Scheduled Period
I Year - 1 Semester | 30/09/2019 - 18/01/2020
Hours: 48 (24 esercitazione, 24 lezione)
Prerequisites
Elementary Calculus and Geometry
Target skills and knowledge
Differential and Symplectic Geometry. Global Hamiltonian Mechanics. Geometric theory of the Hamilton-Jacobi equation. Symplectic Topology. Calculus of Variations: Conjugate Points, Morse Index, Lusternik-Schnirelman Theory for the existence of critical points.
Examination methods
Written.
Assessment criteria
Assessment of learning theoretical and practical notions on the course.
Course contents
Essential of Differential Geometry and Exterior Differential Calculus.
Cohomology.
Riemannian manifolds:
Existence of metrics, Whitney theorem.
Symplectic Geometry:
Symplectic manifolds.
Introduction and developments of Hamiltonian Mechanics on symplectic manifolds.
Local and global parameterization of the Lagrangian submanifolds and their
generating functions. Theorem
of Maslov-H\"ormander.
Hamilton-Jacobi equation, its geometrical solutions and links to the Calculus of
Variations. Conjugate points
theory in calculus of variations.
Relative cohomology and Lusternik-Schnirelman theory. Introduction to Symplectic
Topology: existence and classification of critical points of
functions and applications to generating functions of Lagrangian submanifolds.
The min-max solution of Hamilton-Jacobi equation. Symplectic Topology by Viterbo: towards the solution of the Arnol'd conjecture. Morse theory.
Planned learning activities and teaching methods
lectures and tutorials
Additional notes about suggested reading
F. Cardin: Elementary Symplectic Topology and Mechanics, Springer 2015
Textbooks (and optional supplementary readings)
- Hofer, Helmut; Zehnder, Eduard, Symplectic invariants and Hamiltonian dynamics, Birkhäuser, 1994.
- McDuff, Dusa, Salamon, Dietmar, Introduction to symplectic topology, Oxford Mathematical Monographs, 1998.
- Arnolʹd, V. I., Mathematical methods of classical mechanics, Springer Verlag, 1989.
- F. Cardin, Elementary Symplectic Topology and Mechanics, Springer Verlag, 2015.