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Course unit
SYMPLECTIC MECHANICS
SC02119743, A.A. 2019/20
Information concerning the students who enrolled in A.Y. 2019/20
ECTS: details
Type |
Scientific-Disciplinary 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 |
Examination board
Board |
From |
To |
Members of the board |
8 Meccanica Superiore - a.a. 2019/2020 |
01/10/2019 |
30/09/2020 |
CARDIN
FRANCO
(Presidente)
BERNARDI
OLGA
(Membro Effettivo)
FASSO'
FRANCESCO
(Supplente)
FAVRETTI
MARCO
(Supplente)
GUZZO
MASSIMILIANO
(Supplente)
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Prerequisites:
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Elementary Calculus and Geometry |
Target skills and knowledge:
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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:
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Written. |
Assessment criteria:
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Assessment of learning theoretical and practical notions on the course. |
Course unit contents:
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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:
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lectures and tutorials |
Additional notes about suggested reading:
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F. Cardin: Elementary Symplectic Topology and Mechanics, Springer 2015 |
Textbooks (and optional supplementary readings) |
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Hofer, Helmut; Zehnder, Eduard, Symplectic invariants and Hamiltonian dynamics. --: Birkhäuser, 1994.
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Arnolʹd, V. I., Mathematical methods of classical mechanics. Springer Verlag: 1989, --.
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McDuff, Dusa, Salamon, Dietmar, Introduction to symplectic topology. --: Oxford Mathematical Monographs, 1998.
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F. Cardin, Elementary Symplectic Topology and Mechanics. --: Springer Verlag, 2015.
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