What are Darboux coordinates? What are Darboux coordinates?
Are they different from coordinates on $\Bbb R^n$ or some smooth manifold?
I am familiar with Riemannian manifolds, but Darboux coordinates came up in some materials.

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.
