Editing Symplectic geometry (section)

==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>