Carathéodory–Jacobi–Lie theorem I am studying Analytical Mechanics and want to prove the following theorem:
Let be $(M, \omega)$ a sympletic manifold, $U \subset M $ open,  $f_1,\ \ldots,\ f_n \in C(U)$ such that 
1 $\{f_i,f_j\}=0 \ \forall i,\ j= 1,\ \ldots,\ n$;
2 $\{ f_1,\ \ldots,\ f_n\}$    linearly independent.
So, $\exists g_1,\ \ldots,\ g_n \in C(U)$: 
$$\omega|_U = \sum_{i=1}^{n} dg_i \wedge  {df_i}.$$ 
(Remark: $C(U)=\{ f: U\rightarrow    \mathbb{R},\   C^{\infty}\}$ and $\{f_i,f_j\}$ is the Poisson bracket)
I think this theorem is a consequence of the Darboux theorem, but I did not find the proof of this result in the books that I'm using to study. Can someone indicate a book that has the proof or give me a hint on how to prove this theorem?
 A: The Carathéodory-Jacobi-Lie theorem is kind of a strengthened version of the Darboux theorem, as the latter one only says that on some neighbordhood of any point in a symplectic manifolds there exist functions $f_1, \dots, f_n, g_1, \dots, g_n$ as in your question, whereas the former one starts from any given set of such $f_i$'s. However, in the Darboux theorem, it is required to have $\{ g_i, g_j \} = 0$ too.
Notice that in general, we can only find functions $g_j$'s on a smaller open neighbordhood $V \subset U$ ; The reason why should come out of the proof.

Let $X_i$ be the Hamiltonian vector field on $U$ associated to the function $f_i$. By definition, we have $X_i \, \lrcorner \, \left. \omega \right|_U =  \mathrm{d}f_i$. Notice that $0 = \{ f_i, f_j   \} = \left. \omega \right|_u (X_i, X_j) = \mathrm{d}f_i(X_j)$. If the Carathéodory-Jacobi-Lie theorem were true, we would have
$$ \mathrm{d}f_i = X_i \, \lrcorner \, \left. \omega \right|_U = \sum_{j=1}^n \, \mathrm{d}g_j(X_i) \, \mathrm{d}f_j \, . $$
Since the $f_i$'s are assumed linearly independent, the above equality is possible if and only if $\mathrm{d}g_j(X_i) = \delta_{ij}$ (the Kronecker delta). In particular, $\mathrm{d}g_i(X_i) = 1$ : the function $g_i$ has to be more or less a time parameter associated to the flow of $X_i$. To select 'the' appropiate time parameter, we have to take into account the other relations we obtained.

Notice that $[X_i, X_j]$ is the Hamiltonian vector field associated to $\{ f_i, f_j \}$, hence it is identically zero. The linear independence of the $f_i$'s is equivalent to the linear independence of the $X_i$'s. Hence, the $X_i$'s determine a $n$-dimensional isotropic (i.e. a Lagrangian) involutive distribution. By Frobenius integrability theorem, this distribution is integrable ; So $U$ is foliated by Lagrangian submanifolds $\{L_{\alpha}\}_{\alpha \in A}$ and the $f_i$'s are constant on any of these submanifolds.
Given any $i$, consider also the $(n-1)$-dimensional isotropic distribution generated by the vector fields $X_j$ with $j \neq i$ ; It is also involutive and so integrable. So $U$ is foliated by isotropic submanifolds $\{J^{(i)}_{\beta}\}_{\beta \in B_i}$ and the $f_j$'s ($j \neq i$) are constant on these submanifolds.
Take $o \in U$ and consider some $n$-dimensional submanifold $N$ going through $o$ and everywhere transverse to the Lagrangian leaves. Take $p \in N$. There is only one leaf $L_p$ of the Lagrangian distribution going through $p$, as there is only one leaf $J^{(i)}_p$ of the $i$-th isotropic distribution going through $p$. Of course, the $J^{(i)}_p$'s are submanifolds of $L_p$. For $\epsilon > 0$ small enough, the flow $\phi^t_i$ of $X_i$ is defined on a neighbordhood $V$ of $o$ for all $t \in (- \epsilon, \epsilon)$ ; For an even smaller $\epsilon' > 0$, there is an even smaller neighbordhood $V'$ such that the flow does not map $V'$ outside $V$. If they are all taken small enough, the fundamental theorem of ODE assures that given any $q \in V \cap L_p$, there is one point $q'_i=q'_i(q) \in J^{(i)_p} \cap V'$ and one time $t_i=t_i(q) \in (- \epsilon', \epsilon')$ such that $q = \phi^{t_i}_i(q'_i)$.
We define $g_i : V \to \mathbb{R} : q \mapsto t_i(q)$. Notice that $(\phi^t_i)X_j = X_j$, as the $t$-derivative of the LHS is the Lie derivative $L_{X_i}X_j = [X_i, X_j] = 0$. In particular, the flow $\phi^t_i$ preserves the $i$-th isotropic foliation : leaves are sent to leaves. Since $g_i(\phi^t_i(J^{(i)}_p \cap V'))= t$ is a constant for each $t$, we deduce that $\mathrm{d}g_i(X_j) = \delta_{ij}$ everywhere on $V$. So these functions $g_i$'s are 'the' sought-after ones (they depend on the choice of $N$).
