{{Short description|Closed degenerate differential 2-form of constant rank}} In [[mathematical physics]], especially [[geometric mechanics]], a '''presymplectic form''' is a [[geometric structure]] on [[Differentiable manifold|differentiable manifolds]]. It is a generalization of [[symplectic manifold|symplectic form]].
Given a differentiable manifold, a symplectic form over it is [[Differential form|differential 2-form]] that is [[Closed and exact differential forms|closed]] and [[Nondegenerate quadratic form|nondegenerate]]. A presymplectic form relaxes the requirement for nondegeneracy. Instead, it is merely required to be closed and have constant [[Rank (linear algebra)|rank]] at all points on the manifold.<ref>{{Cite journal|last=Vaisman|first=Izu|title=Geometric quantization on presymplectic manifolds|journal=Monatshefte für Mathematik|language=en|volume=96|issue=4|pages=293–310|doi=10.1007/BF01471212|issn=0026-9255|year=1983|s2cid=123233096}}</ref> Note that a symplectic form, by virtue of nondegeneracy, necessarily have rank equaling the dimension of the underlying manifold, so it has constant rank.<ref name="Martınez2003">{{cite web |last=Martınez |first=Eduardo |title=Symplectic, Presymplectic, Poisson, Dirac, ... |url=http://andres.unizar.es/~wdgmp/EduardoMartinez.pdf |url-status=dead |archive-url=https://web.archive.org/web/20130612123337/http://andres.unizar.es/~wdgmp/EduardoMartinez.pdf |archive-date=12 June 2013 |accessdate=26 July 2013}}</ref>
The definition is not standardized. Recently, Hajduk and Walczak defined a presymplectic form as a closed, differential 2-form, ''of maximal rank on a manifold of odd dimension''.<ref name="Hajduk2010">{{cite arXiv |eprint=0912.2297 |class=math.SG |author=Boguslaw Hajduk |author2=Rafa Walczak |title=Presymplectic manifolds |name-list-style=amp |year=2009}}</ref> This may be motivated thus: A symplectic form necessarily exists over a manifold of even dimension, so a manifold of odd dimension cannot have a symplectic form. However, it can at least attempt to reach a rank as high as possible, since a sympletic form, by virtue of nondegeneracy, necessarily have rank equaling the dimension of the underlying manifold, which is the maximal rank possible on the manifold.
== Applications == Presymplectic forms have been used to study physical systems where there is no obvious symplectic geometry underlying it. Examples include dynamical systems with singular [[Lagrangian mechanics|Lagrangians]], [[Hamiltonian system]]s with [[First class constraint|constraints]], and [[control theory]].<ref name="Alishah2012">{{cite web|last=Alishah|first=Hassan Najafi|title=KAM Theory, Presymplectic Dynamics and Lie algebroids|url=http://www.math.illinois.edu/~ruiloja/Estudantes/TeseHNajafi.pdf|publisher=UNIVERSIDADE TÉCNICA DE LISBOA INSTITUTO SUPERIOR TÉCNICO|accessdate=26 July 2013}}</ref>
==References== {{reflist}}
[[Category:Dynamical systems]] [[Category:Differential geometry]]
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