Editing Symplectic geometry

{{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".

==Overview==
{{quotebox|width=60%|align=right
|quote=The name "complex group" formerly advocated by me in allusion to line complexes, as these are defined by the vanishing of antisymmetric bilinear forms, has become more and more embarrassing through collision with the word "complex" in the connotation of complex number. I therefore propose to replace it by the corresponding Greek adjective "symplectic". Dickson called the group the "Abelian linear group" in homage to Abel who first studied it.
|source={{harvtxt|Weyl|1939|p=165}}}}
A symplectic geometry is defined on a smooth even-dimensional space that is a [[differentiable manifold]]. On this space is defined a geometric object, the [[Symplectic manifold#Definition|symplectic 2-form]], that allows for the measurement of sizes of two-dimensional objects in the [[Space (mathematics)|space]]. The symplectic form in symplectic geometry plays a role analogous to that of the [[metric tensor]] in [[Riemannian geometry]]. Where the metric tensor measures lengths and angles, the symplectic form measures oriented areas.<ref name=McDuff2010>{{citation|last=McDuff|first=Dusa|contribution=What is Symplectic Geometry?|title=European Women in Mathematics – Proceedings of the 13th General Meeting|editor-last=Hobbs|editor-first=Catherine|editor2-last=Paycha|editor2-first=Sylvie|date=2010|publisher=World Scientific|isbn=9789814277686|pages=33–51|citeseerx=10.1.1.433.1953}}</ref>

Symplectic geometry arose from the study of [[classical mechanics]] and an example of a symplectic structure is the motion of an object in one dimension. To specify the trajectory of the object, one requires both the [[Position (geometry)|position]] ''q'' and the [[momentum]] ''p'', which form a point (''p'',''q'') in the [[Two-dimensional Euclidean space|Euclidean plane]] <math>\mathbb{R}^{2}</math>. In this case, the symplectic [[Differential form|form]] is
:<math>\omega = dp \wedge dq</math>
and is an [[Volume form|area form]] that measures the area ''A'' of a region ''S'' in the plane through [[Differential form#Integration|integration]]:
:<math>A = \int_S \omega.</math>
The area is important because as [[Conservative system|conservative dynamical systems]] evolve in time, this area is invariant.<ref name=McDuff2010/>

Higher dimensional symplectic geometries are defined analogously. A 2''n''-dimensional symplectic geometry is formed of pairs of directions
: <math>((x_1,x_2), (x_3,x_4),\ldots(x_{2n-1},x_{2n}))</math>
in a 2''n''-dimensional manifold along with a symplectic form
:<math>\omega = dx_1 \wedge dx_2 + dx_3 \wedge dx_4 + \cdots + dx_{2n-1} \wedge dx_{2n}.</math>
This symplectic form yields the size of a 2''n''-dimensional region ''V'' in the space as the sum of the areas of the projections of ''V'' onto each of the planes formed by the pairs of directions<ref name=McDuff2010/>

:<math>A = \int_V \omega = \int_V dx_1 \wedge dx_2 + \int_V dx_3 \wedge dx_4 + \cdots + \int_V dx_{2n-1} \wedge dx_{2n}.</math>

==Comparison with Riemannian geometry==
Symplectic geometry has a number of similarities with and differences from [[Riemannian geometry]], which is the study of [[differentiable manifold]]s equipped with nondegenerate, symmetric 2-tensors (called [[metric tensor]]s). Unlike in the Riemannian case, symplectic manifolds have no local invariants such as [[curvature of Riemannian manifolds|curvature]]. This is a consequence of [[Darboux's theorem]] which states that a neighborhood of any point of a 2''n''-dimensional symplectic manifold is isomorphic to the standard symplectic structure on an open set of <math>\mathbb{R}^{2n}</math>. Another difference with Riemannian geometry is that not every differentiable manifold need admit a symplectic form; there are certain topological restrictions. For example, every symplectic manifold is even-dimensional and [[orientable]]. Additionally, if ''M'' is a closed symplectic manifold, then the 2nd [[de Rham cohomology]] [[group (mathematics)|group]] ''H''<sup>2</sup>(''M'') is nontrivial; this implies, for example, that the only [[n-sphere|''n''-sphere]] that admits a symplectic form is the [[sphere|2-sphere]]. A parallel that one can draw between the two subjects is the analogy between [[geodesics]] in Riemannian geometry and [[pseudoholomorphic curve]]s  in symplectic geometry: Geodesics are curves of shortest length (locally), while pseudoholomorphic curves are surfaces of minimal area. Both concepts play a fundamental role in their respective disciplines.

==Examples and structures==
Every [[Kähler manifold]] is also a symplectic manifold. Well into the 1970s, symplectic experts were unsure whether any compact non-Kähler symplectic manifolds existed, but since then many examples have been constructed (the first was due to [[William Thurston]]); in particular, [[Robert Gompf]] has shown that every [[finitely presented group]] occurs as the [[fundamental group]] of some symplectic 4-manifold, in marked contrast with the Kähler case.

Most symplectic manifolds, one can say, are not Kähler; and so do not have an integrable [[Linear complex structure|complex structure]]  compatible with the symplectic form. [[Mikhail Gromov (mathematician)|Mikhail Gromov]], however, made the important observation that symplectic manifolds do admit an abundance of compatible [[almost complex structure]]s, so that they satisfy all the axioms for a Kähler manifold ''except'' the requirement that the [[transition map]]s be [[Holomorphic function|holomorphic]].

Gromov used the existence of almost complex structures on symplectic manifolds to develop a theory of [[pseudoholomorphic curve]]s,<ref>Gromov, Mikhael. "Pseudo holomorphic curves in symplectic manifolds." Inventiones mathematicae 82.2 (1985): 307–347.</ref> which has led to a number of advancements in symplectic topology, including a class of symplectic invariants now known as [[Gromov–Witten invariant]]s. Later, using the pseudoholomorphic curve technique [[Andreas Floer]] invented another important tool in symplectic geometry known as the [[Floer homology]].<ref>Floer, Andreas. "Morse theory for Lagrangian intersections." Journal of differential geometry 28.3 (1988): 513–547.</ref>

==See also==
{{Div col|colwidth=25em}}
* [[Contact geometry]]
* [[Geometric mechanics]]
* [[Moment map]]
* [[Poisson geometry]]
* [[Symplectic duality]]
* [[Symplectic integrator|Symplectic integration]]
* [[Symplectic resolution]]
* [[Symplectic vector space]]
{{Div col end}}

==Notes==
{{Reflist}}

==References==
* {{cite book |first1=Ralph |last1=Abraham |author-link=Ralph Abraham (mathematician) |first2=Jerrold E. |last2=Marsden |author-link2=Jerrold E. Marsden |title=Foundations of Mechanics |year=1978 |publisher=Benjamin-Cummings |location=London |isbn=978-0-8053-0102-1 }}
* {{Cite journal |last=Arnol'd |first=V. I. |year=1986 |title=Первые шаги симплектической топологии |trans-title=First steps in symplectic topology |url=http://stacks.iop.org/0036-0279/41/i=6/a=R01?key=crossref.b64c82f40738bc9e5d8af2d5ca203307 |language=ru |journal=Успехи математических наук |volume=41 |issue=6(252) |pages=3–18 |doi=10.1070/RM1986v041n06ABEH004221 |s2cid=250908036 |issn=0036-0279 |via=[[Russian Mathematical Surveys]], 1986, 41:6, 1–21}}
* {{cite book |first1=Dusa |last1=McDuff |author-link=Dusa McDuff |first2=D. |last2=Salamon |author-link2=Dietmar Arno Salamon |title=Introduction to Symplectic Topology |publisher=Oxford University Press |year=1998 |isbn=978-0-19-850451-1 }}
* {{cite book |first=A. T. |last=Fomenko |title=Symplectic Geometry |edition=2nd |year=1995 |publisher=Gordon and Breach |isbn=978-2-88124-901-3 }} ''(An undergraduate level introduction.)''
* {{cite book |first=Maurice A. |last=de Gosson |author-link=Maurice A. de Gosson |title=Symplectic Geometry and Quantum Mechanics |year=2006 |publisher=Birkhäuser Verlag |location=Basel |isbn=978-3-7643-7574-4 }}
* {{cite journal |first=Alan |last=Weinstein |author-link=Alan Weinstein |year=1981 |title=Symplectic Geometry |journal=[[Bulletin of the American Mathematical Society]] |volume=5 |issue=1 |pages=1–13 |url=http://www.ams.org/bull/1981-05-01/S0273-0979-1981-14911-9/S0273-0979-1981-14911-9.pdf |doi=10.1090/s0273-0979-1981-14911-9|doi-access=free }}
* {{Cite book | last1=Weyl | first1=Hermann | author1-link=Hermann Weyl | title=The Classical Groups. Their Invariants and Representations | year = 1939}} Reprinted by [[Princeton University Press]] (1997). {{ISBN|0-691-05756-7}}. {{MR|0000255}}.

==External links==
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*{{springer|title=Symplectic structure|id=p/s091860}}

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[[Category:Symplectic geometry| ]]