User:ReiVaX/GDT

Generalized Darboux's theorem is a theorem in symplectic topology which generalizes the Darboux's theorem.

The statement is as follows. Let M be a 2n-dimensional symplectic manifold with symplectic form ω. Let $f_{1},f_{2},\ldots ,f_{r}(r\leq n)$ functions linearly invariant $(df_{1}(p)\wedge \ldots \wedge df_{r}(p)\neq 0)$ at each point such that {f_i, f_j} = 0 (they are within involution, {-,-} is the Poisson bracket). Then there are functions $f_{i+1},\ldots ,f_{n},g_{1},\ldots ,g_{n}$ such that (f_i, g_i) is a symplectic chart of M, i.e.

$\omega =\sum _{i=1}^{n}df_{i}(p)\wedge dg_{i}(p)$.

Category:Symplectic topology Category:Mathematical theorems