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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.

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 $i,j$.

2 deleted a superfluous appendix

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.

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 $i,j$.

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On the proof of the hamiltonian flow box theorem

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.

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 $i,j$.