
Course unit
SYMPLECTIC MECHANICS
SC02119743, A.A. 2019/20
Information concerning the students who enrolled in A.Y. 2019/20
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Core courses 
MAT/07 
Mathematical Physics 
6.0 
Course unit organization
Period 
First semester 
Year 
1st Year 
Teaching method 
frontal 
Type of hours 
Credits 
Teaching hours 
Hours of Individual study 
Shifts 
Practice 
3.0 
24 
51.0 
No turn 
Lecture 
3.0 
24 
51.0 
No turn 
Prerequisites:

Elementary Calculus and Geometry 
Target skills and knowledge:

Differential and Symplectic Geometry. Global Hamiltonian Mechanics. Geometric theory of the HamiltonJacobi equation. Symplectic Topology. Calculus of Variations: Conjugate Points, Morse Index, LusternikSchnirelman Theory for the existence of critical points. 
Examination methods:

Written. 
Assessment criteria:

Assessment of learning theoretical and practical notions on the course. 
Course unit 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 MaslovH\"ormander.
HamiltonJacobi equation, its geometrical solutions and links to the Calculus of
Variations. Conjugate points
theory in calculus of variations.
Relative cohomology and LusternikSchnirelman theory. Introduction to Symplectic
Topology: existence and classification of critical points of
functions and applications to generating functions of Lagrangian submanifolds.
The minmax solution of HamiltonJacobi 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.

Arnolʹd, V. I., Mathematical methods of classical mechanics. Springer Verlag: 1989, .

McDuff, Dusa, Salamon, Dietmar, Introduction to symplectic topology. : Oxford Mathematical Monographs, 1998.

F. Cardin, Elementary Symplectic Topology and Mechanics. : Springer Verlag, 2015.


