User:ReiVaX/GDT

From Wikipedia, the free encyclopedia

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 {fi, fj} = 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 (fi, gi) 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