PlanetMath: Darboux's theorem (symplectic geometry)

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Darboux's theorem (symplectic geometry)

(Theorem)

If $(M,\omega)$ is a -dimensional symplectic manifold, and $m\in M$ , then there exists a neighborhood of with a coordinate chart

$\displaystyle x=(x_1,\ldots,x_{2n}):U\to\mathbb{R}^{2n},$

such that

$\displaystyle \omega=\sum_{i=1}^ndx_{i}\wedge dx_{n+i}.$

These are called canonical or Darboux coordinates. On

, $\omega$ is the pullback by

of the standard symplectic form on $\mathbb{R}^{2n}$ , so

is a symplectomorphism. Darboux's theorem implies that there are no local invariants in symplectic geometry, unlike in Riemannian geometry, where there is curvature.

"Darboux's theorem (symplectic geometry)" is owned by bwebste.

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Other names:

Darboux coordinates

Attachments:: proof of Darboux's theorem (symplectic geometry) (Proof) by rspuzio

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53D05 (Differential geometry :: Symplectic geometry, contact geometry :: Symplectic manifolds, general)

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