Darboux coordinates

Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among them being symplectic geometry. The theorem is named after Jean Gaston Darboux who established it as the solution of the Pfaff problem.

One of the many consequences of the theorem is that any two symplectic manifolds of the same dimension are locally symplectomorphic to one another. That is, every 2n-dimensional symplectic manifold can be made to look locally like the linear symplectic space Cⁿ with its canonical symplectic form. There is also an analogous consequence of the theorem as applied to contact geometry.

The precise statement is as follows. Suppose that ${\displaystyle \theta }$ is a differential 1-form on an n dimensional manifold, such that ${\displaystyle \mathrm {d} \theta }$ has constant rank p. If

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