I am reading this very popular introduction on symplectic geometry by Cannas. The story goes as follows: Having manifolds $X_1, X_2$ , we want to construct symplectomorphisms between $M_1,M_2$ (where $M_i=T^*X_i$). Now to to that we use to things:
The first is that a closed $1-$form $w$ on the manifold $X$ defines $Χ_w=\{x,w_x\}$, a submanifold of $T^*X$. The fact that $dw=0$ forces $X_w$ to be Lagrangian. This means that for any function $f:X_1\times X_2 \rightarrow \mathbb{R}$ , $X_{df}$ is a lagrangian submanifold.
On the other hands we have that a diffeomorphism $φ:X_1\rightarrow X_2$ is a symplectomoprhism if-f the graph of $φ$ in $(X_1\times X_2,\tilde{ω})$ is lagrangian.
In the notes (page $29$ for example) it is said , or maybe I am misinterpreting something, that every $f:X_1\times X_2 \rightarrow \mathbb{R}$ gives rise to a symplectomorphism, by means of solving the corresponding hamiltonian equations. But I don't see a proof of that in the notes; So my question is : Is it true that the hamiltonian equations are always sovlable, i.e. the lagrangina submanifold $X_{df}$ is always the graph of a diffeomorphism ?