Darboux's theorem

Darboux's theorem is a theorem in symplectic topology which states that every symplectic manifold (of fixed dimension) is locally symplectomorphic. That is, every 2n-dimensional symplectic manifold can be made to look locally like the linear symplectic space Cn with its canonical symplectic form.

Related Topics:
Theorem - Symplectic topology - Symplectic manifold - Symplectomorphic - Linear symplectic space

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The precise statement is as follows. Let M be a 2n-dimensional symplectic manifold with symplectic form ω. Then around every point p in M there exists a coordinate chart U containing p with coordinates (x_1, y_1, x_2, y_2, ldots, x_n, y_n) such that on U, ω is of the form

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:omega = sum_{i=1}^{n} dx^i wedge dy^i.

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Stated differently, if φ : U → Cn is the coordinate chart then ω is the pullback of the standard form ω0 on Cn:

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:omega = phi^{*}omega_0,.

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The chart U is said to be a Darboux chart around p. The manifold M can be covered by such charts. The transition functions in such an atlas will be given by symplectic matrices.

Related Topics:
Covered - Atlas - Symplectic matrices

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