The hamiltonian flow box theorem, as stated in Abraham and Marsden's Foundations of Mechanics, says that:


Given an hamiltonian system $(M,\omega,h)$ with $dh(x\_0)\neq 0$ for some $x\_0$ in $M$, there is a symplectic chart $(U,\phi)$ on $M$ centered at $x\_0$ such that $\phi\_{\ast}h(x)=h(x\_0)+\omega\_0(\phi\_{\ast}X\_h(x\_0),x)$, where $\omega\_0$ is the canonical symplectic form.


\**Question:*\*I know some different proofs of this theorem, but I would know if, at your knowledge, in the literature there is a proof which uses the Moser's trick as in the proof of the Darboux' theorem.




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In Abraham and Marsden there is a proof using the contact structure associated to the symplectic one and its canonical transormations.
I know even that it has an extension in a theorem of Cartan which says:
Given a $2n$-dimensional symplectic manifold $(M,\omega)$, it is possible to extend to a system of symplectic coordinates on $(M,\omega)$ any set of local functions $f\_1,\ldots,f\_k,g\_1,\ldots,g\_l$ on $M$ such that $f\_1,\ldots,f\_k$ are independent and in involution, $g\_1,\ldots,g\_l$ are independent and in involution, and $\{f\_i,g\_j\}=\delta\_{ij}$ for any $i,j$.



On the proof of the hamiltonian flow box theorem
My guess would be that you can find such a proof in the literature, since the Moser trick is such a powerful tool, though I don't know where. 


Instead let me sketch a proof of the fact that any two Hamiltonian systems $(M\_i,\omega\_i,h\_i)$ are locally isomorphic around non deg. points $x\_i\in M\_i, i=0,1$ using the Moser trick. That's the statement I see behind the flow box theorem.


First as usal one proves the linearized fact (i.e. any two $2n$ dim vector spaces $V\_i$ supplied with non-degenerate two-forms $\omega\_i$ and non degenerate one forms $v\_i, i=0,1$ are isomorphic). Using this one constructs a local diffeomorphism between $(M\_0,\omega\_0,h\_0)$ and $(M\_1,\omega\_1,h\_1)$ satisfying:


1. point $x\_0$ goes to $x\_1$
2. the symplectic forms coincide on the above points and
3. the Hamiltonian function $h\_0$ goes to $h\_1$.


Now we have a manifold $M$ with two symplectic forms $\omega\_0,\omega\_1$ coinciding at $x\in M$ and one function $h$ which is nondegenerate at $x$. Next you try the usual Moser trick to morph $\omega\_1$ to $\omega\_2$ with the flow of a time dependent vector field $X\_t$, imposing the additional requirement that $X\_t$ preserve the function $h$. I.e. $L\_{X\_t}h=0$. Hence $X\_t$ should lie in the $2n-1$ dimensional distribution $\ker dh$.


At some point in the Moser trick one chooses a one-form $\alpha$ such that $d\alpha=\omega\_1-\omega\_0$, and here we have the freedom to fullfill the additional restriction since we can add any closed one form $df$ to $\alpha$. We want the result $\alpha+df$ to lie in the $2n-1$ dimensional subspace of one-forms satisfyinig $i\_{Y\_t}\alpha'=0$, where $Y\_t$ denotes the Hamiltonian vector field associated to $h$ w.r.t. the symplectic structure $\omega\_t$. This can always be achieved since $Y\_t$ is non deg and you are solving the equation $Y\_t(f)=g\_t$ where $g\_t=-i\_{Y\_t}\alpha$.


