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Darboux theorem

From Encyclopedia of Mathematics
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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 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, $S_{ij}(z)=\omega(\frac{\partial}{\partial z_i},\frac{\partial}{\partial z_i})$ in any local coordinate system $(z_1,\dots,z_{2n})$ is antisymmetric and nondegenerate: $\omega=\frac12\sum S_{ij}(z)\,\rd z_i\land \rd z_j$.

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 $$ \omega=\sum_{i=1}^n \rd x_i\land \rd p_i. $$

Darboux therem for intermediate values of differentiable functions

How to Cite This Entry:
Darboux theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Darboux_theorem&oldid=25694
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article