# Presymplectic form

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Presymplectic_form
> Markdown URL: https://mediated.wiki/source/Presymplectic_form.md
> Source: https://en.wikipedia.org/wiki/Presymplectic_form
> Source revision: 1346659086
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

Closed degenerate differential 2-form of constant rank

In [mathematical physics](/source/Mathematical_physics), especially [geometric mechanics](/source/Geometric_mechanics), a **presymplectic form** is a [geometric structure](/source/Geometric_structure) on [differentiable manifolds](/source/Differentiable_manifold). It is a generalization of [symplectic form](/source/Symplectic_manifold).

Given a differentiable manifold, a symplectic form over it is [differential 2-form](/source/Differential_form) that is [closed](/source/Closed_and_exact_differential_forms) and [nondegenerate](/source/Nondegenerate_quadratic_form). A presymplectic form relaxes the requirement for nondegeneracy. Instead, it is merely required to be closed and have constant [rank](/source/Rank_(linear_algebra)) at all points on the manifold.[1] Note that a symplectic form, by virtue of nondegeneracy, necessarily have rank equaling the dimension of the underlying manifold, so it has constant rank.[2]

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*.[3] 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 [Lagrangians](/source/Lagrangian_mechanics), [Hamiltonian systems](/source/Hamiltonian_system) with [constraints](/source/First_class_constraint), and [control theory](/source/Control_theory).[4]

## References

1. **[^](#cite_ref-1)** Vaisman, Izu (1983). "Geometric quantization on presymplectic manifolds". *Monatshefte für Mathematik*. **96** (4): 293–310. [doi](/source/Doi_(identifier)):[10.1007/BF01471212](https://doi.org/10.1007%2FBF01471212). [ISSN](/source/ISSN_(identifier)) [0026-9255](https://search.worldcat.org/issn/0026-9255). [S2CID](/source/S2CID_(identifier)) [123233096](https://api.semanticscholar.org/CorpusID:123233096).

1. **[^](#cite_ref-Martınez2003_2-0)** Martınez, Eduardo. ["Symplectic, Presymplectic, Poisson, Dirac, ..."](https://web.archive.org/web/20130612123337/http://andres.unizar.es/~wdgmp/EduardoMartinez.pdf) (PDF). Archived from [the original](http://andres.unizar.es/~wdgmp/EduardoMartinez.pdf) (PDF) on 12 June 2013. Retrieved 26 July 2013.

1. **[^](#cite_ref-Hajduk2010_3-0)** Boguslaw Hajduk & Rafa Walczak (2009). "Presymplectic manifolds". [arXiv](/source/ArXiv_(identifier)):[0912.2297](https://arxiv.org/abs/0912.2297) [[math.SG](https://arxiv.org/archive/math.SG)].

1. **[^](#cite_ref-Alishah2012_4-0)** Alishah, Hassan Najafi. ["KAM Theory, Presymplectic Dynamics and Lie algebroids"](http://www.math.illinois.edu/~ruiloja/Estudantes/TeseHNajafi.pdf) (PDF). UNIVERSIDADE TÉCNICA DE LISBOA INSTITUTO SUPERIOR TÉCNICO. Retrieved 26 July 2013.

This differential geometry–related article is a stub. You can help Wikipedia by adding missing information.

- [v](https://en.wikipedia.org/wiki/Template:Differential-geometry-stub)
- [t](/source/Template_talk%3ADifferential-geometry-stub)
- [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Differential-geometry-stub)

---
Adapted from the Wikipedia article [Presymplectic form](https://en.wikipedia.org/wiki/Presymplectic_form) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Presymplectic_form?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.