Editing Symplectic geometry (section)

{{short description|Branch of differential geometry and differential topology}}

[[File:Limitcycle.svg|thumb|340px|right|[[Phase portrait]] of the [[Van der Pol oscillator]], a one-dimensional system. [[Phase space]] was the original object of study in symplectic geometry.]]

'''Symplectic geometry''' is a branch of [[differential geometry]] and [[differential topology]] that studies [[symplectic manifold]]s; that is, [[differentiable manifold]]s equipped with a [[closed differential form|closed]], [[nondegenerate form|nondegenerate]] [[differential form|2-form]]. Symplectic geometry has its origins in the [[Hamiltonian mechanics|Hamiltonian formulation]] of [[classical mechanics]] where the [[phase space]] of certain classical systems takes on the structure of a symplectic manifold.<ref>{{cite news |first=Kevin |last=Hartnett |date=February 9, 2017 |title=A Fight to Fix Geometry's Foundations |work=[[Quanta Magazine]] |url=https://www.quantamagazine.org/the-fight-to-fix-symplectic-geometry-20170209/ }}</ref>

The term "symplectic", introduced by [[Hermann Weyl]],<ref>Weyl, Hermann (1939). The Classical Groups. Their Invariants and Representations. Reprinted by Princeton University Press (1997). ISBN 0-691-05756-7. MR0000255</ref> is a [[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 comes from the corresponding Greek ''sym-plektikos'' (συμπλεκτικός); in both cases the stem comes from the Indo-European root [[wiktionary:Reconstruction:Proto-Indo-European/pleḱ-|*pleḱ-]]  The name reflects the deep connections between complex and symplectic structures.

By [[Darboux's theorem]], symplectic manifolds are isomorphic to the standard [[symplectic vector space]] locally, hence only have global (topological) invariants. "Symplectic topology," which studies global properties of symplectic manifolds, is often used interchangeably with "symplectic geometry".