What are Darboux coordinates?

Given a symplectic structure $\omega$ on a $(2 m)$-manifold $M$---that is a closed $2$-form such that $d\omega^m \neq 0$---Darboux's Theorem guarantees that for every point $p \in M$ there are local coordinates $(x^i, y^i)$ in which the coordinate representation of $\omega$ is $$ \hat{\omega} = \sum_{i = 1}^m dx^i \wedge dy^i.$$ We call such coordinates Darboux coordinates and sometimes say that they are adapted to the symplectic form. The fact that Darboux coordinates always exist means that for any two symplectic manifolds $(M, \omega)$ and $(M', \omega')$ and points $p \in M$ and $p' \in M'$, there is a local diffeomorphism between neighborhoods of $p$ and $p'$ that pulls back $\omega'$ to $\omega$. In other words, the existence guarantees that all symplectic structures are locally equivalent.

This contrasts sharply with the behavior of Riemannian metrics. Given a Riemannian $n$-manifold $(M, g)$ and a point $p \in M$, in general there are not coordinates $(x^i)$ for which the coordinate representation of $g$ is $\hat g = \sum_{i = 1}^n (dx^i)^2$. In fact, the existence of such coordinates is equivalent to the vanishing of the Riemannian curvature in some neighborhood of $p$, and hence to the existence of a local isometry around $p$ of $(M, g)$ and the flat metric. Thus, a consequence of Darboux's Theorem is that for symplectic structures there is no analogue of curvature---or any local invariant, for that matter.

Another (related) version of Darboux coordinates is instead adapted to a contact structure on a manifold, or more generally, a $1$-form $\theta$ for which $d\theta$ has constant rank.

A smooth manifold $M$ equipped with a closed, non-degenerate two-form $\omega$ is called a symplectic manifold. It follows almost immediately that a symplectic manifold is even dimensional.

Darboux's Theorem: Let $(M, \omega)$ be a symplectic manifold. For any $p \in M$, there is a coordinate chart $(U, (x^1, \dots, x^n, y^1, \dots, y^n))$ with $p \in U$ such that $\omega$ takes the form $$dx^1\wedge dy^1 + \dots + dx^n\wedge dy^n.$$

The coordinates in the above theorem are often called Darboux coordinates.