# Darboux’s theorem (symplectic geometry)

If $(M,\omega )$ is a $2n$-dimensional symplectic manifold^{}, and $m\in M$, then there exists a
neighborhood^{} $U$ of $m$ with a coordinate chart^{}

$$x=({x}_{1},\mathrm{\dots},{x}_{2n}):U\to {\mathbb{R}}^{2n},$$ |

such that

$$\omega =\sum _{i=1}^{n}d{x}_{i}\wedge d{x}_{n+i}.$$ |

These are called canonical or Darboux coordinates. On $U$, $\omega $ is the pullback by $X$
of the standard symplectic form on ${\mathbb{R}}^{2n}$, so $x$ is a symplectomorphism. Darboux’s theorem
implies that there are no local invariants in symplectic geometry, unlike in Riemannian geometry,
where there is curvature^{}.

Title | Darboux’s theorem (symplectic geometry) |
---|---|

Canonical name | DarbouxsTheoremsymplecticGeometry |

Date of creation | 2013-03-22 13:15:31 |

Last modified on | 2013-03-22 13:15:31 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 6 |

Author | bwebste (988) |

Entry type | Theorem |

Classification | msc 53D05 |

Synonym | Darboux coordinates |