Difference between revisions of "Darboux theorem"
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| − | 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\ | + | 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 | 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 | ||
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\omega=\sum_{i=1}^n \rd x_i\land \rd p_i. | \omega=\sum_{i=1}^n \rd x_i\land \rd p_i. | ||
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| + | '''Theorem''' (Darboux theorem<ref name=AG>Arnold V. I., Givental A. B.</ref>, sometimes also referred<ref name=AG/> to as the Darboux-Weinstein theorem<ref>Guillemin V., Sternberg S., </ref>). | ||
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| + | <references/> | ||
== Darboux therem for intermediate values of differentiable functions == | == Darboux therem for intermediate values of differentiable functions == | ||
Revision as of 12:54, 29 April 2012
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, $S_{ij}(z)=\omega(\frac{\partial}{\partial z_i},\frac{\partial}{\partial z_i})=-S_{ji}(z)$ 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 $$ \omega=\sum_{i=1}^n \rd x_i\land \rd p_i. $$
Theorem (Darboux theorem[1], sometimes also referred[1] to as the Darboux-Weinstein theorem[2]).
Darboux therem for intermediate values of differentiable functions
Darboux theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Darboux_theorem&oldid=25693