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 $$x=(x_1,\ldots,x_{2n}):U\to\R^{2n},$$ such that $$\omega=\sum_{i=1}^ndx_{i}\wedge dx_{n+i}.$$ These are called canonical or Darboux coordinates. On $U$ , $\omega$ is the pullback by $X$ of the standard symplectic form on $\R^{2n}$ , 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.