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{{Short description|A closed degenerate differential 2-form of constant rank}} |
{{Short description|A closed degenerate differential 2-form of constant rank}} |
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In geometric mechanics a |
In geometric mechanics a ”’presymplectic form”’ is a ”'[[Closed and exact differential forms|closed]] [[Differential form|differential 2-form]] of constant rank on a [[manifold (mathematics)|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> a of [[symplectic form]] is a [[ form|nondegenerate]].<ref name=”Martınez2003″>{{cite web|last=Martınez|first=Eduardo|title=Symplectic, Presymplectic, Poisson, Dirac, …|url=http://andres.unizar.es/~wdgmp/EduardoMartinez.pdf|= |archive-url=https://web.archive.org/web/20130612123337/http://andres.unizar.es/~wdgmp/EduardoMartinez.pdf|archive-date=12 June 2013|=}}</ref> |
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Lack of nondegeneracy, leading to presymplectic forms, occurs in 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> |
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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> |
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==References== |
==References== |
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Latest revision as of 17:40, 15 October 2025
A closed degenerate differential 2-form of constant rank
In mathematical physics, especially geometric mechanics, a presymplectic form is a closed differential 2-form of constant rank on a manifold.[1] It is a generalization of symplectic form, which is a closed, nondegenerate, differential 2-form.[2]
Lack of nondegeneracy, leading to presymplectic forms, occurs in dynamical systems with singular Lagrangians, Hamiltonian systems with constraints and control theory.[3]
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.[4]
