DarbouxTheorem

The Darboux theorem, named after Jean Gaston Darboux, is a foundational result in symplectic geometry. It asserts that every symplectic manifold is locally indistinguishable from standard Euclidean space with the canonical symplectic form. In more precise terms, if (M, ω) is a symplectic manifold of dimension 2n and p is any point in M, there exists a neighborhood U of p and a coordinate chart φ: U → R^{2n} with coordinates (x1, ..., xn, y1, ..., yn) such that ω|_U = sum_i dxi ∧ dyi. These coordinates are called Darboux coordinates.

Consequences of the theorem include the absence of local symplectic invariants: all symplectic manifolds look locally the same, which contrasts with the local invariants present in Riemannian geometry (such as curvature). The richness of symplectic geometry therefore lies in global, rather than local, properties.

Sketch of the proof: ω is closed and nondegenerate, so by the Poincaré lemma one can locally write ω = dα. A standard approach uses a Moser-type argument to construct a local diffeomorphism that carries ω to the constant standard form sum_i dxi ∧ dyi. Thus, after a suitable coordinate change, ω becomes the standard form in a neighborhood of p. There are several equivalent proofs of this local normal form.

History and context: The theorem is a central result in the development of symplectic geometry in the late 19th and early 20th centuries and has numerous extensions, including analogous statements in contact geometry.