Symplectic geometry
branch of differential geometry and differential topology
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold.
Etymology
The term "symplectic", as adopted into mathematics by Hermann Weyl, is a neo-Greek calque of "complex". Previously, the "symplectic group" had been called the "line complex group". "Complex" comes from the Latin com-plexus, meaning "braided together" (co- + plexus), while "symplectic" represents the corresponding Greek sym-plektikos (συμπλεκτικός "twining or plaiting together, copulative"). In both cases, the stems come from the Indo-European root *pleḱ-, expressing the concept of folding or weaving, and the prefixes suggest "togetherness". The name reflects the deep connections between complex and symplectic structures.
By Darboux's theorem, symplectic manifolds are locally isomorphic to the standard symplectic vector space. Hence they have only global (topological) invariants. The term "symplectic topology" is often used interchangeably with "symplectic geometry".
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This entry incorporates text from “Symplectic geometry” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.