Say we have a symplectic basis $\{e_{1},\ldots,e_{n},f_{1},\ldots,f_{n}\}$ for $(T_{p}M,\omega_{p})$. If $\{e_{1}^{*},\ldots,e_{n}^{*},f_{1}^{*},\ldots,f_{n}^{*}\}$ is the dual basis of $\left(T_{p}M\right)^{*}$, then $\omega_{p}\in\wedge^{2}\left(T_{p}M\right)^{*}$ has the following expression:
$$
\omega_{p}=\sum_{i=1}^{n}e_{i}^{*}\wedge f_{i}^{*}.
$$
So we have to show that there exists a coordinate chart $(x_{1}',\ldots,x_{n}',y_{1}',\ldots,y_{n}')$ around $p$ satisfying
$$
(dx_{i}')_{p}=e_i^{*},\\
(dy_{i}')_{p}=f_{i}^{*}.
$$
Such a chart can be constructed as follows. Let $(\xi_{1},\ldots,\xi_n,\eta_1,\ldots,\eta_{n})$ be any coordinate chart around $p$. Now both $\{e_{1}^{*},\ldots,e_{n}^{*},f_{1}^{*},\ldots,f_{n}^{*}\}$ and $\{(d\xi_{1})_{p},\ldots,(d\xi_{n})_{p},(d\eta_{1})_{p},\ldots,(d\eta_{n})_{p}\}$ are bases of $(T_{p}M)^{*}$. Let the following block matrix be the change of bases:
$$
\begin{bmatrix}
A & C\\ B&E \tag{1}
\end{bmatrix},
$$
i.e.
$$
e_{1}^{*}=\sum_{j=1}^{n}a_{j1}(d\xi_{j})_{p}+\sum_{j=1}^{n}b_{j1}(d\eta_{j})_{p},\\
\vdots\\
e_{n}^{*}=\sum_{j=1}^{n}a_{jn}(d\xi_{j})_{p}+\sum_{j=1}^{n}b_{jn}(d\eta_{j})_{p},\\
f_{1}^{*}=\sum_{j=1}^{n}c_{j1}(d\xi_{j})_{p}+\sum_{j=1}^{n}e_{j1}(d\eta_{j})_{p},\\
\vdots\\
f_{n}^{*}=\sum_{j=1}^{n}c_{jn}(d\xi_{j})_{p}+\sum_{j=1}^{n}e_{jn}(d\eta_{j})_{p}.
$$
This suggests that we should define
$$
x_{1}'=\sum_{j=1}^{n}a_{j1}\xi_{j}+\sum_{j=1}^{n}b_{j1}\eta_{j},\\
\vdots\\
x_{n}'=\sum_{j=1}^{n}a_{jn}\xi_{j}+\sum_{j=1}^{n}b_{jn}\eta_{j},\\
y_{1}'=\sum_{j=1}^{n}c_{j1}\xi_{j}+\sum_{j=1}^{n}e_{j1}\eta_{j},\\
\vdots\\
y_{n}'=\sum_{j=1}^{n}c_{jn}\xi_{j}+\sum_{j=1}^{n}e_{jn}\eta_{j}.
$$
To check that $(x_{1}',\ldots,x_{n}',y_{1}',\ldots,y_{n}')$ are really coordinates around $p$, we should check that the map $$(\xi_{1},\ldots,\xi_n,\eta_1,\ldots,\eta_{n})\mapsto(x_{1}'(\xi,\eta),\ldots,x_{n}'(\xi,\eta),y_{1}'(\xi,\eta),\ldots,y_{n}'(\xi,\eta))$$
is a local diffeomorphism around $p$. This is the case, since its Jacobian is exactly the transpose of the block matrix (1) above, which is non-singular.
I am still unsure what are you after; here are some relevant results:
- Suppose that $(M,\omega_0)$ is a compact symplectic manifold. Consider a smooth perturbation of $\omega_0$, i.e. a smooth family of symplectic forms $\omega_t$, $t\in [0,T]$. One question to ask is if there is a smooth family of diffeomorphisms $f_t: M\to M, t\in [0,T]$, such that $f_0=id_M$ and $f_t^*(\omega_t)=\omega_0$. There is an obvious topological obstruction to the existence of such a family, namely, the cohomology classes $[\omega_t]\in H^2(M, {\mathbb R})$ have to be constant (i.e. the same as the one given by $\omega_0$). In other words, for each $t$ there should be a 1-form $\alpha_t\in \Omega^1(M)$ such that $\omega_t- \omega_0= d\alpha_t$.
Now, the relevant theorem is known as Moser's Stability Theorem:
Theorem 1. Assume that in the above setting $[\omega_t]=[\omega_0]$ for all $t$. Then indeed, there is a smooth family of diffeomorphisms $f_t: M\to M, t\in [0,T]$, such that $f_0=id_M$ and $f_t^*(\omega_t)=\omega_0$.
- Moser's theorem generalizes to noncompact manifolds, for instance:
Theorem 2. Suppose that $(M,\omega_t)$ is a symplectic manifold and $\omega_t$ as above is:
a. A compactly supported deformation of $\omega_0$ in the sense that:
There is a compact $K\subset M$ such that $\omega_0=\omega_t$ outside of $K$ for all $t\in [0,T]$, and the compactly supported cohomology class of $(\omega_t-\omega_0), t\in [0,T]$, is zero.
Then there exists a smooth family of diffeomorphisms $f_t: M\to M, t\in [0,T]$, such that $f_0=id_M$ and $f_t^*(\omega_t)=\omega_0$ and, furthermore, $f_t=id, t\in [0,T]$, outside of a compact subset $C\subset M$.
b. In the case when $\omega_0$ is the standard symplectic form on $M={\mathbb R}^{2n}$ one can do a bit better and find a family of diffeomorphisms $f_t: M\to M, t\in [0,T]$ such that $f_0=id_M$ and $f_t^*(\omega_t)=\omega_0$, provided that the difference $\omega_t(x)-\omega_0(x)$ merely decays sufficiently fast (in a suitable sense) as $x\to \infty$.
One can think of Theorem 2 as a version of the Global Darboux Theorem on ${\mathbb R}^{2n}$ for "small perturbations" of the standard symplectic form.
- One can also ask if the Global Darboux Theorem holds for arbitrary symplectic manifolds $(M,\omega)$. One obvious obstruction, of course is that $M=M^{2n}$ is supposed to be diffeomorphic to a domain in ${\mathbb R}^{2n}$. With this restriction, Global Darboux again holds for planar surfaces ($n=1$), due to Greene and Shiohama, generalizing Moser's proof. However, Global Darboux fails in dimensions $\ge 4$ even if $M={\mathbb R}^{2n}$, $n\ge 2$. This was first observed by Gromov (who left a proof as an exercise as he tends to). Explicit examples were found later, for instance, in works by Bates, Peschke and Casals:
Theorem 3. For every $n\ge 2$ there exists a symplectic form $\omega$ on ${\mathbb R}^{2n}$ such that there is no smooth embedding
$f: {\mathbb R}^{2n}\to {\mathbb R}^{2n}$ satisfying
$$
f^*(\omega_0)= \omega,
$$
where $\omega_0$ is the standard symplectic form on ${\mathbb R}^{2n}$.
References:
Larry Bates, George Peschke, A remarkable symplectic structure, J. Differ. Geom. 32, No. 2, 533-538 (1990). ZBL0714.53028.
Roger Casal, Exotic symplectic structures, ZBL07152607.
Robert Greene, Katsuhiro Shiohama, Diffeomorphisms and volume-preserving embeddings of noncompact manifolds, Trans. Am. Math. Soc. 255, 403-414 (1979). ZBL0418.58002.
Jürgen Moser, On the volume elements on a manifold, Trans. Am. Math. Soc. 120, 286-294 (1965). ZBL0141.19407.
Xiudi Tang, "Symplectic Stability and New Symplectic Invariants of Integrable Systems", Ph.D. thesis, 2018.
See also this lecture by Weimin Chen for a self-contained treatment of Moser's theorem.
Best Answer
Here's a proof along the lines that you are looking for and which is valid only in dimension 2.
Consider a smoothly embedded open ball $B \subset M$. Since $B$ is contractible and the restriction of the symplectic form $\omega$ to $B$ is still closed, this restriction is exact. Thus there exists a 1-form $\lambda$ on $B$ such that $\omega = d \lambda$; by adding to $\lambda$ a closed 1-form and considering if necessary a smaller ball, we can assume without loss of generality that $\lambda$ does not vanish on $B$. From the result mentioned in the question, $\lambda$ can be written as $\lambda = f dg$ for some smooth functions $f, g \in C^{\infty}(B)$. We thus deduce that $\omega = d\lambda = d(fdg) = df \wedge dg$. Hence, considering $\mathbb{R}^2$ equipped with coordinates $(x,y)$ and the standard symplectic form $\omega_0 = dx \wedge dy$, we get that the map $\Phi : B \to \mathbb{R}^2$ given by $\Phi(p) = (f(p), g(p))$ is symplectic, i.e. $\Phi^* \omega_0 = \omega$. It follows that $\Phi$ is an immersion, and it is also a submersion since $B$ and $\mathbb{R}^2$ have the same dimension. This map is thus a local diffeomorphism; Upon shrinking $B$ again if necessary, we can assume that $\Phi$ is a symplectic diffeomorphism onto its image i.e. a Darboux chart.