Home » Second Level Degree » Courses » SYMPLECTIC MECHANICS

Second Level Degree in Mathematics

Jump second level menu

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)

Move Up ▲

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)

Torna su ▲