Darboux theorem

Darboux theorem 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.

For Darboux theorem on integrability of differential equations, see Darboux integral.

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: $\omega=\frac12\sum_{i,j=1}^{2n} S_{ij}(z)\,\rd z_i\land \rd z_j$.

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

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 (1): 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 $h^*\omega_1=\omega_2$.

Relative versions

Together with the "absolute" version, one has a "relative" version of the Darboux theorem: if $M$ is a smooth manifold with two symplectic structures $\omega_1,\omega_2$, and $N$ is a submanifold on which the two 2-forms coincide, then near each point $a\in N\subseteq M$ one has a diffeomorphism $h:(M,a)\to(M,a)$ transforming $\omega_1$ to $\omega_2$ and identical on $N$: $$ \omega_1=\omega_2\Big|_{TN}\ \implies\ \exists h\in\operatorname{Diff}(M,a):\quad h^*\omega_1=\omega_2,\quad h|_N\equiv\operatorname{id}. $$

Comments

The assertion of the Darboux theorem on local normalization of antisymmetric 2-forms should be compared with a similar question about symmetric nondegenerate forms, which (if positive) define a Riemannian metric on $M$. It is well known that, although at a given point $a$ the Riemannian metric can be brought to the canonical form $\left<v,v\right>=\sum_{i=1}^n v_i^2$, such transformation is in general impossible in any open neighborhood of $a$: the obstruction, among other things, is represented by the curvature of the metric (which is zero for the "constant" standard Euclidean metric).

In the same way the relative Darboux theorem means that submaniolds of the symplectic manifold have no "intrinsic" geometry: any two submanifolds $N,N'$ with equivalent (eventually, quite degenerate) restrictions of $\omega$ on $TN$, resp., $TN'$, can be transformed to each other by a diffeomorphism preserving the symplectic structure.

Darboux theorem for intermediate values of differentiable functions

2020 Mathematics Subject Classification: Primary: 26A06 [MSN][ZBL]

If $f:[a,b]\to\R$ is a function which is differentiable at all points of the segment $[a,b]\subseteq\R$ (the right and left derivatives are assumed at the endpoints $a,b$ respectively), then its derivative assumes all intermediate values (i.e., the range of the derivative $f'=\frac{\rd f}{\rd x}:[a,b]\to\R$ is a connected set).

For functions $f\in C^1[a,b]$ whose derivative is continuous, this is a simple consequence of the intermediate value theorem for the derivative. For functions whose derivative exists at all points but is discontinuous, e.g., $f(x)=x^2\sin(1/x)$, $0\ne x\in[-1,1]$, $f(0)=0$, the assertion follows from the Fermat principle (the derivative of a differential function at an extremal point vanishes), applied to a suitable combination $f(x)-\alpha x$. See also the Darboux property.

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