Content

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   Introduction
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   Symplectic Group
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  Symplectic group
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   Marius Souphous Lie
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   Emergence of Symplectic Groups
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   Weyl and the Symplectic group
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   Symplectic Geometry by Siegel
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   Carl Ludwig Siegel
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   Symplectic geometry by Siegel
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   Siegel Modular forms
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   Quadratic forms
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   Analytical interpretation
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   Siegel modular functions of degree n
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   Symplectic geometry 1943
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   Reception of Siegel's paper 1943
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   Siegel's and Hua's parallel works
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   Hua Loo-Keng
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   Hua in Europe
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   Remarks on Siegel's and Hua's parallel work
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   Differential forms
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   Symplectic form and symplectic space
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  Differential forms
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   Élie Cartan
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   Cartan differential form
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   Closed differential forms
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   Early applications of differential forms
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   Kähler form - Symplectic form
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   Erich Kähler
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   Kähler Metric
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   Reception of Kähler manifolds
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   Eckmann and Guggenheimer
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   Eckmann and Guggenheimer's work on Kähler manifolds
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   Symplectic Manifolds
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   Ehresmann and Feldbau
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   Jacques Feldbau
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   Symplectic Manifolds
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   Fibre Spaces, Fibre Bundle
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   Almost complexes structures
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   Ehresmann: Symplectic Manifolds
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   The first reception and the relation between symplectic manifolds and Kähler manifolds
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   Siegel's Half Space is a Symplectic Manifold
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   Darboux's theorem
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  Two versions of Darboux's theorem
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   The form used in symplectic geometry
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   The classical form
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  A look back to the late 19th century
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   Pfaffian's Problem
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   Gaston Darboux
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   The classical theorem of Darboux
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  The perspective of symplectic manifolds
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   Paulette Libermann
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   Equivalence between symplectic structures
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   About the name of Darboux's Theorem
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   Symplectic Geometry-Lee Hwa-Chung
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   About Lee Hwa-Chung Live
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   Even dimensional geometry
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   Even dimensional manifolds endowed with a nonsingular closed 2-form (Symplectic manifold)
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   Conformal flat manifolds
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   Automorphisms
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   Poisson bracket
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   Hamilton's equations
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   Equivalence of flat spaces
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   Lee Hwa Chung Theorem
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   The reception of Lee's work
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   Classical Mechanics
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  Classical Mechanics
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   Lagrangian and Hamiltonian Mechanics
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   Symplectic Group and Classical Mechanics
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   Aurel Wintner and Canonical Transformations
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   Aurel Wintner
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   Canonical Transformations
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   Lectures on Celestial Mechanics
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  Georges Reeb Systémes Dynamiques
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   George Reeb
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   Systèmes Dynamiques
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   Colloque de Géométrie Différentielle
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  Basic mathematical concepts
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   Lagrangian subspace
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   Submanifolds of a Symplectic Manifold
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   About the Name: Lagrangian Subspaces
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   Symplectomorphisms
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   Strasbourg 1953
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  Géométrie symplectique différentielle
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   Jean-Marie Souriau
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   Differential Symplectic Geometry and applications
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   Vector Space and Symplectic Vector Space
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   Phase space
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   Géométrie Symplectique Différentielle
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    Les Transformations Canoniques - Sympletomorphisms
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   Applications
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  Take off
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   Symplectic geometry takes off
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   Foundations of mechanics
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   Symplectic geometry conferences in 1973 and 1974
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   Textbooks and journals about symplectic geometry
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   Conclusion
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   Appendices
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   Shiing-Shen Chern
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   Chern and Erich Kähler
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   Chern and Élie Cartan
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   Gauss-Bonnet Theorem
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   Vladimir Arnold
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   Arnold's conjecture
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   Symplectic topology
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   Liste of participants
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   Geometria simplettica
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   Géométrie Symplectique
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   Bibliography