Does every Hamiltonian equation can be written in a some system of coordinates as a canonical Hamiltonian system i.e. $\exists(q,p)$ such that $$\begin{equation*} \begin{cases}\dfrac{dq}{dt}=\partial_p \mathcal H\\[0.3cm] \dfrac{dp}{dt}=-\partial_q \mathcal H\end{cases} \end{equation*}$$
And if yes, this is always the case even if the solution of the equation belongs to an infinite dimension space ?