Darboux theorem

Darboux theorem may may refer to one of the following assertions:

• Darboux theorem on local canonical coordinates for symplectic structure;
• Darboux theorem on intermediate values of the derivative of a function of one variable.

Darboux theorems for symplectic structure =

2020 Mathematics Subject Classification: Primary: 37Jxx,53Dxx [MSN][ZBL]

Recall that a symplectic structure on an even-dimensional manifold $M^{2n}$ is a closed nondegenerate $C^\infty$-smooth differential 2-form $\omega$: $$ \omega\in\varLambda^2(M),\qquad \rd \omega=0,\qquad \forall v\in T_p M\quad \exists w\in T_p M:\ \omega_p(v,w)\ne0. $$

The matrix $S(z)$ of a symplectic structure, $S_{ij}(z)=\omega(\frac{\partial}{\partial z_i},\frac{\partial}{\partial z_i})=-S_{ji}(z)$ in any local coordinate system $(z_1,\dots,z_{2n})$ is antisymmetric and nondegenerate: $\omega=\frac12\sum_{1}^{2n} S_{ij}(z)\,\rd z_i\land \rd z_j$.

The standard symplectic structure on $\R^{2n}$ in the standard canonical coordinates $(x_1,\dots,x_n,p_1,\dots,p_n)$ is given by the form $$ \omega=\sum_{i=1}^n \rd x_i\land \rd p_i.\tag* $$ ===Local equivalence Theorem (Darboux theorem, sometimes also referred to as the Darboux-Weinstein theorem). Any symplectic structure locally is $C^\infty$-equivalent to the standard to the standard syplectic structure (*): for any point $a\in M$ there exists a neighborhood $M\supseteq U\owns a$ and "canonical" coordinate functions $(x,p):(U,a)\to (\R^{2n},0)$, such that in these coordinates $\omega$ takes the form $\sum \rd x_i\land\rd p_i$.

In particular, any two symplectic structures $\omega_1,\omega_2$ on $M$ are locally equivalent near each point: there exists the germ of a diffeomorphism $h:(M,a)\to(M,a)$ such that $h^*\omega_1=\omega_2$.