Consider an infinite-dimensional phase space $(X,\omega)$, where $X = V \times V'$ with $V$ being a Banach space and $\omega$ a (weak) symplectic form. Let $E : X \to \mathbb{R}$ be a smooth function, with $E' : X \to X'$ being the derivative. We want to study the Hamiltonian dynamics, i.e., make sense of the weak ODE for $y : [0,T] \to X$, $y(0) = y_0$, $$ \omega(x,\dot{y}) = E'(y)(x),\quad \forall x \in X $$ Clearly, since $E'(x) \in X'$ we cannot in general expect $\dot{y} \in X$, i.e., there are restrictions on $y_0$.
My question: What can one in general say about the existence and uniqueness of the solution $y(t)$? What are the concepts that are needed to describe the solution? What are some good textbooks/reviews covering this type of problems?