Darboux's Theorem

This post will examine Darboux's theorem, an important theorem in symplectic geometry.

Introduction

Darboux's theorem is often stated as follows:

Theorem (Darboux). Let \((M, \omega)\) be a \(2n\)-dimensional symplectic manifold. For every point \(p \in M\), there exist smooth coordinates \((x^1, \dots, x^n, y^1, \dots, y^n)\) centered at \(p\) such that the coordinate representation of the symplectic form \(\omega\) is expressed as follows:

\[\omega = \sum_{i=1}^n dx^i \wedge dy^i.\]

Coordinates satisfying this relation are called Darboux coordinates or symplectic coordinates or canonical coordinates. Similarly, the corresponding chart is called a Darboux chart or symplectic chart or canonical chart, etc.

This theorem can be interpreted in a variety of ways:

• Every symplectic manifold is locally equivalent to the standard symplectic manifold \((\mathbb{R}^{2n}, \bar{\omega})\).
• All symplectic manifolds are locally equivalent.
• There are no local invariants of symplectic manifolds.
• Every symplectic manifold is locally equivalent to the cotangent space of a manifold endowed with the canonical symplectic form.

Symplectomorphisms

Two symplectic manifolds are equivalent if they are symplectomorphic.

Definition (Symplectomorphism). A symplectomorphism is a diffeomorphism \(F : M_1 \rightarrow M_2\) between symplectic manifolds \((M_1, \omega_1)\) and \((M_2, \omega_2)\) such that \(F^*\omega_2 = \omega_1\). If there exists a symplectomorphism between two symplectic manifolds, they are said to be symplectomorphic.

The standard symplectic manifold consists of \((\mathbb{R}^{2n})\) with standard coordinates and the standard symplectic form.

Definition (Standard Symplectic Manifold). The standard symplectic manifold consists of \(\mathbb{R}^{2n}\) together with standard coordinates denoted as \((\bar{x}^1, \dots, \bar{x}^n, \bar{y}^1, \dots, \bar{y}^n)\) and the standard symplectic form \(\bar{\omega}\) defined in terms of standard coordinates as follows:

\[\bar{\omega} = \sum_{i=1}^n d\bar{x}^i \wedge d\bar{y}^i.\]

Thus, all symplectic manifolds possess coordinates such that the coordinate expression of their respective symplectic form is formally similar to the coordinate expression of the standard symplectic form on \(\mathbb{R}^{2n}\). Given any symplectic manifold \((M, \omega)\) and smooth chart \(\varphi = (x^1, \dots, x^n, y^1, \dots, y^n)\) on an open subset \(U \subseteq M\), since the pullback \(\varphi^*\bar{\omega}\rvert_U\) is computed by substitution of the component functions of \(\varphi\) (i.e., the coordinate functions \((x^1, \dots, x^n, y^1, \dots, y^n)\)) for the coordinate functions of the codomain of the map (i.e., \((\bar{x}^1, \dots, \bar{x}^n, \bar{y}^1, \dots, \bar{y}^n)\)), it follows that Darboux's theorem states for \(\bar{U} = \varphi(U)\) that

\begin{align*}\varphi^*\bar{\omega}\rvert_{\bar{U}} &= \varphi^*\left(\sum_{i=1}^n d\bar{x}^i \wedge d\bar{y}^i\right)\rvert_{\bar{U}} \\&= \sum_{i=1}^n dx^i \wedge dy^i \\&= \omega\rvert_U.\end{align*}

To confirm this, we explicitly calculate as follows for any \(p \in M\) and \(v,w \in T_pM\):

\begin{align*}(\varphi^*\bar{\omega}\rvert_{\bar{U}})_p(v, w) &= \bar{\omega}_{\varphi(p)}(d\varphi_p(v), d\varphi_p(w)) \\&= \left(\sum_{i=1}^n d\bar{x}^i\rvert_{\varphi(p)} \wedge d\bar{y}^i\rvert_{\varphi(p)}\right)(d\varphi_p(v), d\varphi_p(w)) \\&= \left(\sum_{i=1}^n d\bar{x}^i\rvert_{\varphi(p)} \otimes d\bar{y}^i\rvert_{\varphi(p)} - d\bar{y}^i\rvert_{\varphi(p)} \otimes d\bar{x}^i\rvert_{\varphi(p)}\right) (d\varphi_p(v), d\varphi_p(w)) \\&= \sum_{i=1}^n \left(d\bar{x}^i\rvert_{\varphi(p)}(d\varphi_p(v)) \cdot d\bar{y}^i\rvert_{\varphi(p)}(d\varphi_p(w)) - d\bar{y}^i\rvert_{\varphi(p)}(d\varphi_p(v)) \cdot d\bar{x}^i\rvert_{\varphi(p)}(d\varphi_p(w)) \right)\\&= \sum_{i=1}^n \left(v(\bar{x}^i \circ \varphi)w(\bar{y}^i \circ \varphi) - v(\bar{y}^i \circ \varphi)w(\bar{x}^i \circ \varphi)\right) \\&= \sum_{i=1}^n \left(v(x^i)w(y^i) - v(y^i)w(x^i)\right) \\&= \sum_{i=1}^n \left(dx^i\rvert_p(v) \cdot dy^i\rvert_p(w) - dy^i\rvert_p(v) \cdot dx^i\rvert_p(w)\right) \\&= \left(\sum_{i=1}^n dx^i\rvert_p \otimes dy^i\rvert_p - dy^i\rvert_p \otimes dx^i\rvert_p\right)(v, w) \\&= \left(\sum_{i=1}^n dx^i\rvert_p \wedge dy^i\rvert_p\right)(v, w) \\&= \omega_p(v, w)\rvert_U.\end{align*}

Thus, we can express Darboux's theorem as follows:

Theorem (Darboux). Let \((M, \omega)\) be a \(2n\)-dimensional symplectic manifold. For every point \(p \in M\), there exists a smooth coordinate chart \((U, \varphi)\) centered at \(p\) such that

\[\varphi^*\bar{\omega}\rvert_{\bar{U}} = \omega\rvert_U,\]

where \((\mathbb{R}^{2n}, \bar{\omega})\) is the standard symplectic manifold and \(\bar{U} = \varphi(U)\).

Definition (Locally Symplectomorphic Manifolds). Symplectic manifolds \((M_1, \omega_1)\) and \((M_2, \omega_2)\) are locally symplectomorphic if around each point \(p \in M_1\) there exists an open neighborhood \(U_p\) and a map \(\varphi_p : U_p \rightarrow V_p\) onto an open subset \(V_p\) of \(M_2\) such that \(\varphi_p\) is a symplectomorphism, i.e., \(\varphi_p\) is a diffeomorphism such that \(\varphi_p^*(\omega_2\rvert_{V_p}) = \omega_1\rvert_{U_p}\).

Thus, Darboux's theorem is equivalent to the statement that every symplectic manifold is locally symplectomorphic to the standard symplectic manifold (i.e., when using the previous definition).

Since the property of being locally symplectomorphic is symmetric and transitive, this also implies that all symplectic manifolds are locally symplectomorphic to each other.

If a local invariant is defined as a property of a symplectic manifold that can distinguish it locally from other symplectic manifolds, then there can be no such local invariants of symplectic manifolds.

Local Equivalence with Cotangent Bundles

The cotangent bundle \(T^*M\) naturally carries a symplectic structure. To demonstrate this, we first define a \(1\)-form \(\tau \in \Omega^1(T^*M)\) called the tautological \(1\)-form. Each element of \(T^*M\) is pair \((q, \varphi)\) consisting of a point \(q \in M\) and a covector \(\varphi \in T_q^*M\). Thus, at each point \((q, \varphi)\) we want to define \(\tau_{(q, \varphi)} \in T_{(q, \varphi)}^*(T^*M)\). The natural projection \(\pi : T^*M \rightarrow M\) is the map

\[\pi(q, \varphi) = q.\]

The point-wise pullback along natural projection is a map with signature \(d\pi^*_{(q, \varphi)} : T^*_qM \rightarrow T_{(q, \varphi)}^*(T^*M)\). We can thus define \(\tau\) as

\[\tau_{(q, \varphi)} = d\pi^*_{(q, \varphi)}\varphi.\]

Thus, \(\tau_{(q, \varphi)}\) pulls \(\varphi\) back along the natural projection at the point \(q\). The action of \(\tau_{(q, \varphi)}\) on a tangent vector \(v \in T_{(q, \varphi)}(T^*M)\) is thus

\[\tau_{(q, \varphi)}(v) = \varphi\left(d\pi_{(q, \varphi)}(v)\right).\]

We can then define a symplectic form \(\omega\) on \(T^*M\) as

\[\omega = -d\tau.\]

We then consider the representation of \(\tau\) in terms of the natural coordinates for \(T^*M\). If we let \((x^i)\) denote smooth coordinates for some open subset of \(M\) and write \(\varphi = \xi_i(q, \varphi) dx^i\rvert_q\) so that the natural coordinates for \(T^*M\) at a point \((q, \varphi)\) become

\[(x^1(\pi(q,\varphi)), \dots, x^n(\pi(q,\varphi)), \xi_1(q, \varphi), \dots, \xi_n(q,\varphi)),\]

we may then compute

\begin{align*}\tau_{(q, \varphi)} &= d\pi^*_{(q, \varphi)}\varphi \\&= d\pi^*_{(q, \varphi)}(\xi_i(q, \varphi) dx^i\rvert_q) \\&= \xi_i(q, \varphi) d\pi^*_{(q, \varphi)}( dx^i\rvert_q) \\&= \xi_i(q, \varphi) d(x^i \circ \pi)\rvert_{(q, \varphi)}.\end{align*}

Denoting the natural coordinate functions \((y^j)\) on \(T^*M\) as

\[(y^1, \dots, y^n, y^{n+1}, \dots, y^{2n}) = (x^1 \circ \pi, \dots, x^n \circ \pi, \xi_1, \dots, \xi_n),\]

the ultimate equation is justified by the following calculation

\begin{align*}d\pi^*_{(q, \varphi)}(dx^i\rvert_q) &= d\pi^*_{(q, \varphi)}(dx^i\rvert_q) \left( \frac{\partial}{\partial y^j}\bigg\rvert_{(q, \varphi)} \right) dy^j\rvert_{(q, \varphi)} \\&= dx^i\rvert_q\left( d\pi_{(q, \varphi)} \left( \frac{\partial}{\partial y^j}\bigg\rvert_{(q, \varphi)} \right)\right)dy^j\rvert_{(q, \varphi)} \\&= dx^i\rvert_q\left( d\pi_{(q, \varphi)} \left( \frac{\partial}{\partial y^j}\bigg\rvert_{(q, \varphi)} \right)(x^k\rvert_q)\frac{\partial}{\partial x^k}\bigg\rvert_q\right)dy^j_{(q, \varphi)} \\&= dx^i\rvert_q\left(\frac{\partial \pi^k}{\partial y^j} \bigg\rvert_{(q, \varphi)}\frac{\partial}{\partial x^k}\bigg\rvert_q\right)dy^j\rvert_{(q, \varphi)}\\&= dx^i\left( \delta^k_j \frac{\partial}{\partial x^k} \right)dy^j\rvert_{(q, \varphi)} \\&= dx^i\left( \frac{\partial}{\partial x^j} \right)dy^j\rvert_{(q, \varphi)} \\&= \delta^i_j dy^j\rvert_{(q, \varphi)} \\&= d(x^i \circ \pi)\rvert_{(q, \varphi)}.\end{align*}

It thus follows that

\[\tau_{(q, \varphi)} = \xi_i(q, \varphi) d(x^i \circ \pi)\rvert_{(q, \varphi)}\]

and thus

\[\tau = \xi_i d(x^i \circ \pi).\]

It is customary to conflate the coordinate functions \(x^j \circ \pi\) on \(T^*M\) with the coordinate functions \(x^j\) on \(M\). In this case, one simply writes

\[\tau = \xi_i dx^i,\]

and the exterior derivative yields

\begin{align*}\omega &= -d\tau \\&= -\left(\sum_i d\xi_i \wedge dx^i\right) \\&= \sum_i dx^i \wedge d\xi_i.\end{align*}

Thus, the coordinate expression for \(\omega\) takes the form of Darboux coordinates. Every cotangent bundle therefore has a canonical symplectic form whose coordinate expression has the form of Darboux coordinates. In particular, each cotangent bundle \(T^*M\) for an \(n\)-dimensional manifold \(M\) is symplectomorphic to the standard symplectic manifold \(\mathbb{R}^{2n}\), which means that every \(2n\)-dimensional symplectic manifold is locally equivalent to a cotangent bundle (particularly, the cotangent bundle \(T^*\mathbb{R}^n\)).

First Proof: Canonical Form of Vector Fields

We will begin with a straightforward inductive proof of Darboux's theorem which relies on the following theorem.

Theorem (Canonical Form of Commuting Vector Fields). Let \(M\) be an \(n\)-dimensional smooth manifold, at let \((V_1,\dots,V_k)\) be a (point-wise) linearly independent \(k\)-tuple of smooth commuting vector fields on an open subset \(W \subseteq M\). For each \(p \in W\), there exists a smooth coordinate chart \((U, (u^i))\) centered at \(p\) such that \(V_i = \partial/\partial u^i\) for \(i = 1,\dots,k\).

In other words, we can arrange for new coordinates such that each of the \(V_i\) are coordinate vector fields \(\partial/\partial u^i\). Such coordinates are canonical insofar as they are expressed in terms of the standard coordinate vector fields. In particular, we can use this theorem to provide a link between Hamiltonian vector fields and coordinate vector fields.

We first establish the following proposition which expresses Darboux coordinates in terms of Poisson brackets.

Proposition. Let \((M, \omega)\) be a \(2n\)-dimensional symplectic manifold. Smooth coordinates \((x^1, \dots, x^n, y^1, \dots, y^n)\) defined on an open subset \(U \subseteq M\) are Darboux coordinates if and only if the following conditions are satisfied with respect to the Poisson bracket:

• \(\{x^i, y^j\} = \delta^{ij}\),
• \(\{x^i, x^j\} = 0\),
• \(\{y^i, y^j\} = 0\).

Proof. Suppose that these conditions are satisfied with respect to the Poisson bracket.

Observe that

\begin{align*}dx^j\left(X_{x^i}\right) &= \omega\left(X_{x^j}, X_{x^i}\right) \\&= 0 \\&= dx^j\left(-\frac{\partial}{\partial y^i}\right)\end{align*}

and

\begin{align*}dy^j\left(X_{x^i}\right) &= \omega\left(X_{y^j}, X_{x^i}\right) \\&= -\omega\left(X_{x^i}, X_{y^j}\right) \\&= -\delta^{ij} \\&= dy^j\left(-\frac{\partial}{\partial y^i}\right).\end{align*}

Thus, since their values under the dual basis covector fields are the same, this implies that \(X_{x^i} = -(\partial/\partial y^i)\). Likewise, \(X_{y^i} = (\partial/\partial x^i)\).

The coordinate expression for \(\omega\) is as follows:

\begin{align*}\omega &= \omega\left(\frac{\partial}{\partial x^i}, \frac{\partial}{\partial x^j}\right) dx^i \otimes dx^j \\&+ \omega\left(\frac{\partial}{\partial y^i}, \frac{\partial}{\partial y^j}\right) dy^i \otimes dy^j \\&+ \omega\left(\frac{\partial}{\partial x^i}, \frac{\partial}{\partial y^j}\right) dx^i \otimes dy^j \\&+ \omega\left(\frac{\partial}{\partial y^i}, \frac{\partial}{\partial x^j}\right) dy^i \otimes dx^j \\&= \omega\left(X_{x^i}, X_{x^j}\right) dx^i \otimes dx^j \\&+ \omega\left(X_{y^i}, X_{y^i}\right) dy^i \otimes dy^j \\&+ \omega\left(X_{x^i}, X_{y^j}\right) dx^i \otimes dy^j \\&+ \omega\left(X_{y^i}, X_{x^j}\right) dy^i \otimes dy^j \\&= 0 + 0 + \delta^{ij}dx^i \otimes dy^j - \delta^{ij}dy^i \otimes dx^j \\&= \sum_{i=1}^n (dx^i \otimes dy^i - dy^i \otimes dx^i) \\&= \sum_{i=1}^n 2 \cdot \mathrm{Alt}(dx^i \otimes dy^i) \\&= \sum_{i=1}^n \frac{(1 + 1)!}{1!1!} \cdot \mathrm{Alt}(dx^i \otimes dy^i) \\&= \sum_{i=1}^n dx^i \wedge dy^i\end{align*}

Conversely, suppose that the coordinates are Darboux coordinates. Using the expression for the Poisson bracket in Darboux coordinates, we compute

\begin{align*}\{x^i, x^j\} &= \sum_{k=1}^n \left(\frac{\partial x^i}{\partial x^k}\frac{\partial x^j}{\partial y^k} - \frac{\partial x^i}{\partial y^k}\frac{\partial x^j}{\partial y^k}\right) \\&= 0\end{align*}

Likewise, we compute that \(\{y^i, y^j\} = 0\). Finally, we compute

\begin{align*}\{x^i, y^j\} &= \sum_{k=1}^n \left(\frac{\partial x^i}{\partial x^k}\frac{\partial y^j}{\partial y^k} - \frac{\partial x^i}{\partial y^k}\frac{\partial y^j}{\partial y^k}\right) \\&= \delta^{ij}.\end{align*}

\(\square\)

We can use this proposition together with the theorem regarding the canonical form of commuting vector fields to prove Darboux's theorem.

Theorem (Darboux). Let \((M, \omega)\) be a symplectic manifold of dimension \(2n\). For every point \(p \in M\) there exist smooth coordinates \((x^1, \dots, x^n, y^1, \dots, y^n)\) centered at \(p\) in which \(\omega\) has the following coordinate representation:

\[\omega = \sum_{i=1}^n dx^i \wedge dy^i.\]

Proof. We will prove the following statement by induction on \(k\): for each \(k = 1,\dots,n\), there exist smooth functions \((x^1, \dots, x^k,y^1, \dots, y^k)\) vanishing at \(p\) such that the tuple of \(1\)-forms \((dx^1, \dots, dx^k, dy^1, \dots, dy^k)\) is (point-wise) linearly independent and such that the following relations are satisfied in a neighborhood of \(p\):

• \(\{x^i, y^j\} = \delta^{ij}\),
• \(\{x^i, x^j\} = 0\),
• \(\{y^i, y^j\} = 0\).

The theorem is established when \(k=n\).

When \(k=1\), choose any smooth function \(x^1\) such that \(dx^1\rvert_p \ne 0\) and \(x^1(p) = 0\). By the theorem regarding the canonical form of commuting vector fields, since \([X_{x^1}, X_{x^1}] = 0\), it follows that there exist smooth coordinates \((u^1,\dots,u^{2n})\) centered at \(p\) such that

\[\frac{\partial}{\partial u^1} = X_{x^1}.\]

Define \(y^1 = -u^1\). Then, as required,

\begin{align*}\{x^1, y^1\} &= \omega(X_{x^1}, X_{y^1}) \\&= -\omega(X_{y^1}, X_{x^1}) \\&= -dy^1(X_{x^1}) \\&= -X_{x^1}(y^1) \\&= -\frac{\partial}{\partial u^1}(-u^1) \\&= 1.\end{align*}

Moreover, note that the tuple \((dx^1, dy^1)\) is (point-wise) linearly independent, since, if \(a dx^1\rvert_p + b dy^1\rvert_p = 0\) at any point \(p\) for some \(a,b \in \mathbb{R}\), then

\begin{align*}0 &= (a dx^1\rvert_p + b dy^1\rvert_p)(X_{x^1}\rvert_p) \\&= a dx^1\rvert_p(X_{x^1}\rvert_p) + b dy^1\rvert_p(X_{x^1}\rvert_p) \\&= 0 + b d(-u)\rvert_p\left(\frac{\partial}{\partial u^1}\bigg\rvert_p\right) \\&= -b\end{align*}

and

\begin{align*}0 &= (a dx^1\rvert_p + b dy^1\rvert_p)(X_{y^1}\rvert_p) \\&= a dx^1\rvert_p(X_{y^1}\rvert_p) + b dy^1\rvert_p(X_{y^1}\rvert_p) \\&= a \omega_p(X_{x^1}\rvert_p, X_{y^1}\rvert_p) + 0 \\&= -a \omega_p(X_{y^1}\rvert_p, X_{x^1}\rvert_p) \\&= -a dy^1\rvert_p(X_{x^1}\rvert_p) \\&=-a d(-u^1)\rvert_p\left(\frac{\partial}{\partial u^1}\bigg\rvert_p\right) \\&= a\end{align*}

and thus \(a = b = 0\).

Now, suppose that, for an arbitrary \(k\) with \(1 \le k \lt n\), we have found smooth functions \((x^1, \dots, x^k,y^1, \dots, y^k)\) satisfying the required condition.

The tuple of vector fields \((X_{x^1}, \dots, X_{x^k}, X_{y^1}, \dots, X_{y^k})\) is point-wise linearly independent. To see this, note that, for any tuple of real-valued coefficients \((a_1, \dots, a_k, b_1, \dots, b_k)\) and any point \(p \in M\), if

\[a_1 X_{x^1}\rvert_p + \dots + a_k X_{x^k}\rvert_p + b_1 X_{y^1}\rvert_p + \dots + b_k X_{y^k}\rvert_p = 0\]

then, by the bilinearity of \(\omega\rvert_p\),

\[a_1 \widehat{\omega}\rvert_p(X_{x^1}\rvert_p) + \dots + a_k \widehat{\omega}\rvert_p(X_{x^k}\rvert_p) + b_1 \widehat{\omega}\rvert_p(X_{y^1}\rvert_p) + \dots + b_k \widehat{\omega}\rvert_p(X_{y^k}\rvert_p) = \widehat{\omega}\rvert_p(0)\]

where \(\widehat{\omega}\rvert_p(v) = v \lrcorner \omega\rvert_p\) for all tangent vectors \(v\) at \(p\) which means that

\[a_1 dx^1\rvert_p + \dots + a_k dx^k\rvert_p + b_1 dy^1\rvert_p + \dots + b_k dy^k\rvert_p = 0,\]

and, since the tuple \((dx^1, \dots, dx^k, dy^1, \dots, dy^k)\) is linearly independent by the inductive hypothesis, it follows that \(a^i = b^i = 0\) for all \(i\).

The vector fields \((X_{x^i}, \dots, X_{x^k}, X_{y^i}, \dots, X_{y^k})\) commute, since

• \([X_{x^i}, X_{x^j}] = -X_{\{x^i, x^j\}} = -X_0 = 0,\)
• \([X_{y^i}, X_{y^j}] = -X_{\{y^i, y^j\}} = -X_0 = 0,\)
• \([X_{x^i}, X_{y^j}] = -X_{\{x^i, y^j\}} = -X_{\delta^{ij}} = 0,\)

where \(\widehat{\omega}(X_{\delta^{ij}}) = d\delta^{ij} = 0\) since \(\delta^{ij}\) is constant which further implies that \(X_{\delta^{ij}} = 0\) by the non-degeneracy of \(\omega\).

Thus, we may apply the theorem regarding canonical forms of commuting vector fields above to produce smooth coordinates \((u^1, \dots, u^{2n})\) vanishing at \(p\) such that

• \(\frac{\partial}{\partial u^i} = X_{x^i}\) for \(i = 1,\dots,k\),
• \(\frac{\partial}{\partial u^{i+k}} = X_{y^i}\) for \(i = 1,\dots,k\).

We then define \(y^{k+1} = u^{2k+1}\).

Note that, for \(i = 1,\dots,k\),

\begin{align*}\{x^i, y^{k+1}\} &= -\{y^{k+1}, x^i\} \\&= -\omega(X_{y^{k+1}}, X_{x^i}) \\&= -\omega(X_{u^{2k+1}}, X_{x^i}) \\&= -du^{2k+1}(X_{x^i}) \\&= -du^{2k+1}\left(\frac{\partial}{\partial u^i}\right) \\&= -\frac{\partial}{\partial u^i}(u^{2k+1}) \\&= 0\end{align*}

and

\begin{align*}\{y^i, y^{k+1}\} &= -\{y^{k+1}, y^i\} \\&= -\omega(X_{y^{k+1}}, X_{y^i}) \\&= -\omega(X_{u^{2k+1}}, \frac{\partial}{\partial u^{i+k}}) \\&= -du^{2k+1}(\frac{\partial}{\partial u^{i+k}}) \\&= -\frac{\partial}{\partial u^{i+k}}(u^{2k+1}) \\&= 0.\end{align*}

In any Poisson bracket, \(\{y^{k+1}, y^{k+1}\} = 0\). Thus, the corresponding vector fields commute, since, for \(i = 1,\dots, k\),

• \([X_{x^i}, X_{y^{k+1}}] = -X_{\{x^i, y^{k+1}\}} = -X_0 = 0,\)
• \([X_{y^i}, X_{y^{k+1}}] = -X_{\{y^i, y^{k+1}\}} = -X_0 = 0,\)
• \([X_{y^{k+1}}, X_{y^{k+1}}] = 0.\)

We may then apply the same theorem a second time to produce new smooth coordinates \((v^1, \dots, v^{2n})\) such that

• \(\frac{\partial}{\partial v^i} = X_{x^i}\) for \(i = 1,\dots,k\),
• \(\frac{\partial}{\partial v^{i+k}} = X_{y^i}\) for \(i = 1,\dots,k+1\).

We then define \(x^{k+1} = v^{2k + 1}\).

Note that, for \(i = 1,\dots,k\),

\begin{align*}\{x^{k+1}, x^i\} &= \omega(X_{x^{k+1}}, X_{x^i}) \\&= \omega(X_{v^{2k+1}}, X_{x^i}) \\&= dv^{2k+1}(X_{x^i}) \\&= dv^{2k+1}(\frac{\partial}{\partial v^i}) \\&= \frac{\partial}{\partial v^i}(v^{2k+1}) \\&= 0.\end{align*}

and for \(i = 1,\dots,k+1\),

\begin{align*}\{x^{k+1}, y^i\} &= \omega(X_{x^{k+1}}, X_{y^i}) \\&= \omega(X_{v^{2k+1}}, X_{y^i}) \\&= dv^{2k+1}(X_{y^i}) \\&= dv^{2k+1}(\frac{\partial}{\partial v^{i+k}}) \\&= \frac{\partial}{\partial v^{i+k}}(v^{2k+1}) \\&= \delta^{2k+1, i+k}.\end{align*}

These coordinates thus satisfy the following additional relations for \(i=1,\dots,k+1\):

• \(\{x^{k+1}, x^i\} = \{x^i, x^{k+1}\} = 0\),
• \(\{x^{k+1}, y^i\} = \delta^{2k+1, i+k}\).

Thus, the coordinates \((x^1, \dots, x^{k+1}, y^1, \dots, y^{k+1})\) satisfy the required relations for \(i,j = 0,\dots,k+1\):

• \(\{x^i, x^j\} = 0\),
• \(\{y^i, y^j\} = 0\),
• \(\{x^i, y^j\} = \delta^{ij}\).

\(\square\)

This proof also reveals an implicit algorithm which is a symplectic analog of the Gram-Schmidt algorithm. The canonical coordinates at each step can be constructed using the composition of the flows of the individual vector fields involved. The previous proof thus represents an implicit iterative procedure for constructing symplectic coordinates starting from a suitable choice for an initial coordinate function.

Second Proof: Moser's Trick

This section will explore a technique commonly known as Moser's trick due to mathematician Jürgen Moser [Moser, 65] and later expanded by Alan Weinstein [Weinstein, 71]. The proof detailed here follows [Lee, Smooth Manifolds], but with much more explicit detail.

Motivation

The basic idea is to deform a symplectic form \(\omega_0\) on \(\mathbb{R}^{2n}\) into the standard symplectic form \(\omega_1 = \sum_{i=1}^n dx^i \wedge dy^i\) on \(\mathbb{R}^{2n}\) via a parameterized family of closed \(2\)-forms \(\omega_t = (1-t)\omega_0 + t\omega_1\). This is accomplished with the aid of a homotopy of diffeomorphisms \(\theta_t : U_0 \rightarrow U_1\) between open subsets \(U_0\) and \(U_1\) of \(\mathbb{R}^{2n}\) for \(t \in [0, 1]\), which is also an isotopy since each map \(\theta_t\) is a diffeomorphism. This isotopy additionally satisfies the condition \(\theta_t^*\omega_t = \omega_0\) for all \(t\), and hence, in particular, \(\theta_1^*\omega_1 = \omega_0\). Thus, given a symplectic form \(\omega\) on a smooth manifold \(M\) and a smooth coordinate chart \((U, \psi)\) with coordinate map \(\psi : U \rightarrow U_0\), if we choose \(\omega_0\) such that \(\omega_0 = (\psi^{-1})^*\omega\), then we obtain

\begin{align*}(\theta_1 \circ \psi)^*\omega_1 &= \psi^*(\theta_1^*\omega_1) \\&= \psi^*\omega_0 \\&= \psi^*((\psi^{-1})^* \omega) \\&= \omega.\end{align*}

Thus, \((U, \varphi)\) where \(\varphi = \theta_1 \circ \psi\) is the requisite chart since \(\varphi^*\omega_1 = \omega\). The key idea is to consider a parametric family of vector fields \(V_t\), i.e., a time-dependent vector field, and to identify the desired diffeomorphisms \(\theta_t\) with the diffeomorphisms induced by the flow \(\theta\) of this time-dependent vector field at some fixed \(t\) (namely \(t=0\), hence \(\theta_t\) denotes \(\theta_{t,0}\)). Thus, the goal is to solve the following equation for all \(t\):

\[\theta^*_{t,0}\omega_t = \omega_0.\]

This is done by differentiating the equation with respect to \(t\) to produce the equation

\[\frac{d}{dt}\bigg\rvert_{t=t_1}(\theta^*_{t,0}\omega_t) = 0.\]

This derivative can be computed using the following theorem, which relates derivatives of pullbacks of smooth time-dependent tensor fields along smooth time-dependent flows to the Lie derivative.

Theorem. For any smooth manifold \(M\), open interval \(J \subseteq \mathbb{R}\), smooth time-dependent vector field \(V : J \times M\) with flow \(\theta : \mathcal{E} \rightarrow M\), smooth time-dependent tensor field \(A : J \times M \rightarrow T^kT^*M\), and tuple \((t_1, t_0, p) \in \mathcal{E}\),

\[\frac{d}{dt}\bigg\rvert_{t=t_1}(\theta_{t,t_0}^*A_t)_p = \left[\theta_{t_1,t_0}^*\left(\mathcal{L}_{V_{t_1}}A_{t_1} + \frac{d}{dt}\bigg\rvert_{t=t_1}A_t\right)\right]_p.\]

It is worthwhile to note the meaning of such a derivative. For a fixed \(t_0\) and \(p\), the map \((\theta_{t,t_0}^*A_t)_p\) as a function of \(t\) alone has signature \((\theta_{t,t_0}^*A_t)_p : \mathbb{R} \rightarrow T^kT_p^*M\) and thus is a map between finite-dimensional normed vector spaces. Thus, the respective derivative is a derivative in the sense of the partial (and hence directional) derivative of a function between finite-dimensional normed vector spaces. Since all norms on finite-dimensional vector spaces are equivalent, the norm is immaterial.

Using the preceding theorem, and also using Cartan's magic formula and the fact that \(\omega_{t_1}\) is closed and thus \(d\omega_{t_1} = 0\), we compute the following equations:

\begin{align*}\frac{d}{dt}\bigg\rvert_{t=t_1}(\theta^*_{t,0}\omega_t) &= \theta^*_{t_1,0}\left(\mathcal{L}_{V_{t_1}}\omega_{t_1} + \frac{d}{dt}\bigg\rvert_{t=t_1}\omega_t\right) \\&= \theta^*_{t_1,0}\left(V_{t_1} \lrcorner d\omega_{t_1} + d(V_{t_1} \lrcorner \omega_{t_1}) + \frac{d}{dt}\bigg\rvert_{t=t_1}\omega_t\right) \\&= \theta^*_{t_1,0}\left(d(V_{t_1} \lrcorner \omega_{t_1}) + \frac{d}{dt}\bigg\rvert_{t=t_1}\omega_t\right).\end{align*}

Now, if we define the time-dependent vector field \(V_t\) such that \(V_t \lrcorner \omega_t = \alpha\) for some smooth \(1\)-form \(\alpha\), we then obtain

\[\frac{d}{dt}\bigg\rvert_{t=t_1}(\theta^*_{t,0}\omega_t) = \theta^*_{t_1,0}\left(d\alpha + \frac{d}{dt}\bigg\rvert_{t=t_1}\omega_t\right).\]

If we can arrange this choice such that

\[\frac{d}{dt}\bigg\rvert_{t=t_1}\omega_t = -d\alpha,\]

then the equation is solved. If \(\omega_t = \omega_0 + t\eta\) for some smooth \(2\)-form \(\eta\), then

\[\frac{d}{dt}\bigg\rvert_{t=t_1}\omega_t = \eta\]

and hence

\[\theta^*_{t_1,0}\left(d\alpha + \frac{d}{dt}\bigg\rvert_{t=t_1}\omega_t\right) = \theta^*_{t_1,0}\left(d\alpha + \eta\right),\]

and thus if \(-\eta = d\alpha\), the equation is solved. By the Poincaré lemma, we can indeed find a smooth \(1\)-form \(\alpha\) such that \(d\alpha = -\eta\). Thus, we just need to ensure that \(\omega_t = \omega_0 + t\eta\). If we set \(\eta = \omega_1 - \omega_0\), we obtain a closed \(2\)-form and

\begin{align*}\omega_t &= \omega_0 + t \eta \\&= \omega_0 + t(\omega_1 - \omega_0) \\&= (1-t)\omega_0 + t\omega_1.\end{align*}

Thus, \(\omega_t\) is a smooth linear parameterization which starts at \(t=0\) with \(\omega_0\) and ends at \(t=1\) with \(\omega_1\). Furthermore, \(\theta_{t,0}^*\omega_t = \omega_0\) for all \(t\), and hence \(\theta_{1,0}^*\omega_1 = \omega_0\).

We now proceed to the proof of Darboux's theorem using Moser's trick.

Proof

Theorem (Darboux). Let \((M, \omega)\) be a symplectic manifold of dimension \(2n\). For every point \(p \in M\) there exist smooth coordinates \((x^1, \dots, x^n, y^1, \dots, y^n)\) centered at \(p\) in which \(\omega\) has the following coordinate representation:

\[\omega = \sum_{i=1}^n dx^i \wedge dy^i.\]

Proof. Let \((M,\omega)\) denote any \(2n\)-dimensional symplectic manifold and let \(p \in M\) be any point whatsoever. Let \((D, \psi)\) be any smooth coordinate chart such that \(p \in D\). We may assume, without loss of generality, that \(\psi(p) = 0\), since coordinate maps may be translated by a constant vector of \(\hat{p} = \psi(p)\) if necessary.

Let \(B_r(\hat{p})\) denote the open ball of radius \(r \in \mathbb{R}\) centered at the point \(\hat{p} = \psi(p)\). We then define the restricted coordinate chart \((U, \psi\rvert_U)\) where \(U = \psi^{-1}(B_r(\hat{p}))\) is a subset of \(M\) which is guaranteed to be open by definition of the topology on \(M\) induced by its respective atlas (which, in particular, ensures that \(U\) is open since \(\psi(U \cap D) = \psi(U)\) is open in \(\mathbb{R}^{2n}\)).

Let \(\omega_0\) denote any symplectic form on \(B_r(\hat{p})\) and let \(\omega_1 = \sum_{i=1}^n dx^i \wedge dy^i\) denote the canonical symplectic form on \(\mathbb{R}^{2n}\). We may assume, without loss of generality, that \(\omega_0\rvert_{\hat{p}} = \omega_1\rvert_{\hat{p}}\), since all symplectic vector spaces (including \(\Lambda^2(T^*_p)(\mathbb{R}^{2n})\)) can be put into canonical form by a linear change of coordinates.

Let \(\eta = \omega_1 - \omega_0\). Then \(\eta\) is a closed \(2\)-form and the Poincaré lemma guarantees that there exists a smooth \(1\)-form \(\alpha\) on \(B_r(\hat{p})\) such that \(d\alpha = -\eta\). We may assume, without loss of generality, that \(\alpha_{\hat{p}} = 0\) since we can subtract a constant-coefficient \(1\)-form from \(\alpha\) if necessary.

Define a \(2\)-form \(\omega_t\) on \(B_r(\hat{p})\) for each \(t \in \mathbb{R}\) as follows:

\[\omega_t = (1-t) \omega_0 + t\omega_1.\]

This \(2\)-form is closed for all \(t\) since it is a linear combination of closed \(2\)-forms. Let \(J\) be a bounded open interval such that \([0,1] \subset J\). Note that

\begin{align*}\omega_t\rvert_{\hat{p}} &= (1-t)\omega_0\rvert_{\hat{p}} + t\omega_1\rvert_{\hat{p}} \\&= (1-t)\omega_0\rvert_{\hat{p}} + t\omega_0\rvert_{\hat{p}} \\&= \omega_0\rvert_{\hat{p}}.\end{align*}

Since \(\omega_0\rvert_{\hat{p}}\) is non-denegerate by definition, so also is \(\omega_t\rvert_{\hat{p}}\) for all \(t\).

In fact, we may find some neighborhood \(U_1\) of \(\hat{p}\) such that \(\omega_t\) is non-degenerate on \(U_1\) for all \(t \in \bar{J}\). To see this, consider the following compactness argument. Consider the continuous map \(f : B_r(\hat{p}) \times \bar{J} \rightarrow \mathbb{R}\) defined as

\[f(q, t) = \mathrm{det}((\omega_t\rvert_q)_{ij}),\]

where \((\omega_t\rvert_q)_{ij}\) is the matrix representation of \(\omega_t\rvert_q\) in terms of any fixed basis. Recall that \(\omega_t\rvert_q\) is non-degenerate if and only if \(\mathrm{det}((\omega_t\rvert_q)_{ij}) \ne 0\). Let \(d = \mathrm{det}((\omega_t)\rvert_{\hat{p}})_{ij}\). Then, by the definition of continuity, for any open neighborhood of \(d\) in \(\mathbb{R}\), say \(B_d(d)\), there exists an open neighborhood \(O_t\) of \((\hat{p}, t)\) such that \(f(O_t) \subseteq B_d(d)\), which implies, by the definition of the product topology on \(B_r(\hat{p}) \times \bar{J}\), that there is a basic open neighborhood \(N_t \times I_t\) containing \((\hat{p}, t)\) for some open set \(N_t \subseteq B_r(\hat{p})\) containing \(\hat{p}\) and open set \(I_t \subseteq \bar{J}\) containing \(t\) such that \(f(N_t \times I_t) \subseteq B_d(d)\) which implies that \(\mathrm{det}((\omega_s\rvert_{\hat{p}})_{ij}) \ne 0\) for all \(q \in N_t\) and \(s \in I_t\), i.e., that \(\omega_s\) is non-degenerate on \(N_t\) for all \(s \in I_t\).

This results in an infinite collection of pairs of open subsets \(N_t\) and \(I_t\). The family of intervals \(\{I_t\}_{t \in \bar{J}}\) covers \(\bar{J}\). By the compactness of the interval \(\bar{J}\), we may replace the infinite cover \(\{I_t\}_{t \in \bar{J}}\) with a finite sub-cover \((I_{t_1},\dots,I_{t_k})\) and take the intersection of this finite sub-cover to define an open set \(U_1 \subseteq B_r(\hat{p})\) on which \(\omega_t\) is non-degenerate for all \(t\), i.e.,

\[U_1 = \bigcap_k N_{t_k}.\]

Since \(\omega_t\) is non-degenerate on \(U_1\), there exists an induced isomorphism \(\widehat{\omega}_t : TU_1 \rightarrow T^*U_1\) defined by \(\widehat{\omega}_t(X) = X \lrcorner \omega_t\) for every \(t \in \bar{J}\). Thus, we may apply this isomorphism to the smooth \(1\)-form \(\alpha\) to define a smooth time-dependent vector field \(V_t : J \times U_1 \rightarrow TU_1\) as follows:

\[V_t \lrcorner \omega_t = \alpha.\]

Thus, since \(\alpha_{\hat{p}} = 0\), it follows that \(V_t\rvert_{\hat{p}} = 0\) also.

Let \(\theta : \mathcal{E} \rightarrow U_1\) denote the time-dependent flow of \(V\) where \(\mathcal{E} \subseteq J \times J \times U_1\) is the respective time-dependent flow domain. By the fundamental theorem on time-dependent flows, it follows that \(\theta(t, 0, \hat{p}) = \hat{p}\) for all \(t \in J\) (i.e., \(\theta_{t,0}\) is the unique maximal integral curve of \(V\) with initial condition \(\theta_{t,0}(\hat{p}) = \hat{p}\)). Thus, the flow domain consists of at least the subset \(J \times \{0\} \times \{\hat{p}\}\).

In fact, we can find a neighborhood \(U_0\) of \(\hat{p}\) such that \([0, 1] \times \{0\} \times U_0 \subseteq \mathcal{E}\) for all \(t \in [0, 1]\). We will use a compactness argument similar to the one previously employed. Since \(\theta\) is continuous, for any open neighborhood \(\tilde{U}_t\) of \(\theta(t, 0, \hat{p})\), the set \(O_t = \theta^{-1}(\tilde{U})_t\) is an open neighborhood of \((t, 0, \hat{p})\), and thus, by the definition of the product topology, there exists some basic open neighborhood \(I_t \times N_t\) such that \(t \in I_t\) and \(\hat{p} \in N_t\). The infinite family of open sets \(\{I_t\}_{t \in [0,1]}\) covers \([0,1]\), and thus, by the compactness of the interval \([0,1]\), there exists a finite sub-cover \((I_{t_1},\dots,I_{t_k})\). We then define \(U_0\) as follows:

\[U_0 = \bigcap_k N_{t_k}.\]

Now, by the aforementioned theorem regarding derivatives of pullbacks of symplectic forms along time-dependent flows, for every \(t \in [0,1]\), it follows that

\begin{align*}\frac{d}{dt}\bigg\rvert_{t=t_1}(\theta^*_{t,0}\omega_t)_{\hat{p}} &= \left[\theta^*_{t_1,0}\left(\mathcal{L}_{V_{t_1}}\omega_{t_1} + \frac{d}{dt}\bigg\rvert_{t=t_1}\omega_t\right)\right]_{\hat{p}} \\&= \left[\theta^*_{t_1,0}\left(V_{t_1} \lrcorner d\omega_{t_1} + d(V_{t_1} \lrcorner \omega_{t_1}) + \eta\right)\right]_{\hat{p}} \\&= \left[\theta^*_{t_1,0}\left(d\alpha + \eta\right)\right]_{\hat{p}} \\&= 0.\end{align*}

For any \(t_1 \in [0, 1]\), by the fundamental theorem of calculus, it follows for any \(v,w \in T_{\hat{p}}U_0\), that

\begin{align*}0 &= \int_0^{t_1}\frac{d}{dt}(\theta_{t,0}^*\omega_t)_{\hat{p}}(v,w)~dt \\&= (\theta_{t_1,0}^*\omega_{t_1})_{\hat{p}}(v,w) - (\theta_{0,0}^*\omega_0)_{\hat{p}}(v,w),\end{align*}

and thus \(\theta_{t_1,0}^*\omega_{t_1} = \theta_{0,0}^*\omega_0 = \omega_0\). In particular, \(\theta_{1,0}^*\omega_1 = \omega_0\). By the fundamental theorem on time-dependent flows, \(\theta_{1,0}\) is a diffeomorphism onto its image and is thus a smooth coordinate map.

If we choose \(\omega_0\) such that \(\omega_0 = (\psi\rvert_U^{-1})^*\omega\), then we obtain

\begin{align*}(\theta_{1,0} \circ \psi\rvert_U)^*\omega_1 &= \psi\rvert_U^*(\theta_{1,0}^*\omega_1) \\&= \psi\rvert_U^*\omega_0 \\&= \psi\rvert_U^*((\psi\rvert_U^{-1})^* \omega) \\&= \omega.\end{align*}

Thus, \((U, \varphi)\) where \(\varphi = \theta_{1,0} \circ \psi\rvert_U\) is the requisite smooth chart centered at \(p\) since \(\varphi^*\omega_1 = \omega\). \(\square\)

References

• [Moser, 1965] Moser, Jürgen: On the volume elements on a manifold. Trans. Am. Math. Soc. 120, 286-294 (1965).
• [Weinstein, 1971] Weinstein, Alan: Symplectic manifolds and their Lagrangian submanifolds. Adv. Math. 6, 329-346 (1971).
• [Lee, Smooth Manifolds] J. M. Lee, Introduction to Smooth Manifolds, 2nd ed., Grad. Texts in Math. 218, Springer, New York, 2013.