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{{About|Darboux's theorem in symplectic geometry|Darboux's theorem related to the [[intermediate value theorem]]|Darboux's theorem (analysis)}} | |||
'''Darboux's theorem''' is a [[theorem]] in the [[mathematics|mathematical]] field of [[differential geometry]] and more specifically [[differential forms]], partially generalizing the [[Frobenius integration theorem]]. It is a foundational result in several fields, the chief among them being [[symplectic geometry]]. The theorem is named after [[Jean Gaston Darboux]]<ref>Darboux (1882).</ref> who established it as the solution of the [[Johann Friedrich Pfaff|Pfaff]] problem.<ref>Pfaff (1814–1815).</ref> | |||
One of the many consequences of the theorem is that any two [[symplectic manifold]]s of the same dimension are locally [[symplectomorphic]] to one another. That is, every 2''n''-dimensional symplectic manifold can be made to look locally like the [[linear symplectic space]] '''C'''<sup>''n''</sup> with its canonical symplectic form. There is also an analogous consequence of the theorem as applied to [[contact geometry]]. | |||
==Statement and first consequences== | |||
The precise statement is as follows.<ref>Sternberg (1964) p. 140–141.</ref> Suppose that θ is a differential 1-form on an ''n'' dimensional manifold, such that dθ has constant [[Exterior_algebra#Rank_of_a_k-vector|rank]] ''p''. If | |||
: θ ∧ (dθ)<sup>p</sup> = 0 everywhere, | |||
then there is a local system of coordinates ''x''<sub>1</sub>,...,''x''<sub>''n''-''p''</sub>, ''y''<sub>1</sub>, ..., ''y''<sub>''p''</sub> in which | |||
: θ = ''x''<sub>1</sub> d''y''<sub>1</sub> + ... + ''x''<sub>''p''</sub> d''y''<sub>''p''</sub>. | |||
If, on the other hand, | |||
: θ ∧ (dθ)<sup>p</sup> ≠ 0 everywhere, | |||
then there is a local system of coordinates ''x''<sub>1</sub>,...,''x''<sub>''n''-''p''</sub>, ''y''<sub>1</sub>, ..., ''y''<sub>''p''</sub> in which | |||
: θ = ''x''<sub>1</sub> d''y''<sub>1</sub> + ... + ''x''<sub>''p''</sub> d''y''<sub>''p''</sub> + d''x''<sub>''p''+1</sub>. | |||
In particular, suppose that ω is a symplectic 2-form on an ''n''=2''m'' dimensional manifold ''M''. In a neighborhood of each point ''p'' of ''M'', by the [[Poincaré lemma]], there is a 1-form θ with dθ=ω. Moreover, θ satisfies the first set of hypotheses in Darboux's theorem, and so locally there is a [[coordinate chart]] ''U'' near ''p'' in which | |||
: θ = ''x''<sub>1</sub> d''y''<sub>1</sub> + ... + ''x''<sub>''m''</sub> d''y''<sub>''m''</sub>. | |||
Taking an [[exterior derivative]] now shows | |||
: ω = dθ = d''x''<sub>1</sub> ∧ d''y''<sub>1</sub> + ... + d''x''<sub>''m''</sub> ∧ d''y''<sub>''m''</sub>. | |||
The chart ''U'' is said to be a '''Darboux chart''' around ''p''.<ref>Cf. with McDuff and Salamon (1998) p. 96.</ref> The manifold ''M'' can be [[cover (topology)|covered]] by such charts. | |||
To state this differently, identify '''R'''<sup>2m</sup> with '''C'''<sup>m</sup> by letting ''z''<sub>j</sub> = ''x''<sub>j</sub> + i ''y''<sub>j</sub>. If φ : ''U'' → '''C'''<sup>''n''</sup> is a Darboux chart, then ω is the [[pullback (differential geometry)|pullback]] of the standard symplectic form ω<sub>0</sub> on '''C'''<sup>''n''</sup>: | |||
:<math>\omega = \phi^{*}\omega_0.\,</math> | |||
==Comparison with Riemannian geometry== | |||
This result implies that there are no local invariants in symplectic geometry: a [[Darboux frame|Darboux basis]] can always be taken, valid near any given point. This is in marked contrast to the situation in [[Riemannian geometry]] where the [[curvature of Riemannian manifolds|curvature]] is a local invariant, an obstruction to the [[metric tensor|metric]] being locally a sum of squares of coordinate differentials. | |||
The difference is that Darboux's theorem states that ω can be made to take the standard form in an ''entire neighborhood'' around ''p''. In Riemannian geometry, the metric can always be made to take the standard form ''at'' any given point, but not always in a neighborhood around that point. | |||
==See also== | |||
*[[Carathéodory-Jacobi-Lie theorem]], a generalization of this theorem. | |||
==Notes== | |||
<references/> | |||
==References== | |||
* {{cite journal|last = Darboux|first = Gaston| title = Sur le problème de Pfaff|journal=Bull. Sci. Math.|year=1882|volume=6|pages=14–36, 49–68|url=http://gallica.bnf.fr/ark:/12148/bpt6k68005v}} | |||
* {{cite journal|last = Pfaff|first = Johann Friedrich|title = Methodus generalis, aequationes differentiarum partialium nec non aequationes differentiales vulgates, ultrasque primi ordinis, inter quotcunque variables, complete integrandi|journal = Abhandlungen der Königlichen Akademie der Wissenschaften in Berlin|year=1814–1815|pages=76–136}} | |||
* {{cite book|last=Sternberg|title=Lectures on Differential Geometry|first=Shlomo|publisher=Prentice Hall|year=1964}} | |||
* {{cite book|author=McDuff, D. and Salamon, D.|title=Introduction to Symplectic Topology|publisher=Oxford University Press|year=1998|isbn=0-19-850451-9}} | |||
==External links== | |||
* {{planetmath reference|id=5589|title=Proof of Darboux's Theorem}} | |||
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/darboux_-_pfaff_problem_i.pdf G. Darboux, "On the Pfaff Problem," transl. by D. H. Delphenich] | |||
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/darboux_-_pfaff_problem_ii.pdf G. Darboux, "On the Pfaff Problem (cont.)," transl. by D. H. Delphenich] | |||
{{DEFAULTSORT:Darboux's Theorem}} | |||
[[Category:Differential systems]] | |||
[[Category:Symplectic geometry]] | |||
[[Category:Coordinate systems in differential geometry]] | |||
[[Category:Theorems in differential geometry]] | |||
[[Category:Mathematical physics]] | |||
Revision as of 03:01, 28 February 2013
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Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among them being symplectic geometry. The theorem is named after Jean Gaston Darboux[1] who established it as the solution of the Pfaff problem.[2]
One of the many consequences of the theorem is that any two symplectic manifolds of the same dimension are locally symplectomorphic to one another. That is, every 2n-dimensional symplectic manifold can be made to look locally like the linear symplectic space Cn with its canonical symplectic form. There is also an analogous consequence of the theorem as applied to contact geometry.
Statement and first consequences
The precise statement is as follows.[3] Suppose that θ is a differential 1-form on an n dimensional manifold, such that dθ has constant rank p. If
- θ ∧ (dθ)p = 0 everywhere,
then there is a local system of coordinates x1,...,xn-p, y1, ..., yp in which
- θ = x1 dy1 + ... + xp dyp.
If, on the other hand,
- θ ∧ (dθ)p ≠ 0 everywhere,
then there is a local system of coordinates x1,...,xn-p, y1, ..., yp in which
- θ = x1 dy1 + ... + xp dyp + dxp+1.
In particular, suppose that ω is a symplectic 2-form on an n=2m dimensional manifold M. In a neighborhood of each point p of M, by the Poincaré lemma, there is a 1-form θ with dθ=ω. Moreover, θ satisfies the first set of hypotheses in Darboux's theorem, and so locally there is a coordinate chart U near p in which
- θ = x1 dy1 + ... + xm dym.
Taking an exterior derivative now shows
- ω = dθ = dx1 ∧ dy1 + ... + dxm ∧ dym.
The chart U is said to be a Darboux chart around p.[4] The manifold M can be covered by such charts.
To state this differently, identify R2m with Cm by letting zj = xj + i yj. If φ : U → Cn is a Darboux chart, then ω is the pullback of the standard symplectic form ω0 on Cn:
Comparison with Riemannian geometry
This result implies that there are no local invariants in symplectic geometry: a Darboux basis can always be taken, valid near any given point. This is in marked contrast to the situation in Riemannian geometry where the curvature is a local invariant, an obstruction to the metric being locally a sum of squares of coordinate differentials.
The difference is that Darboux's theorem states that ω can be made to take the standard form in an entire neighborhood around p. In Riemannian geometry, the metric can always be made to take the standard form at any given point, but not always in a neighborhood around that point.
See also
- Carathéodory-Jacobi-Lie theorem, a generalization of this theorem.
Notes
References
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Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
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