In this paper, Toms discusses the method that Faddeev and Jackiw proposed for quantization of constrained theories. In section III.B, he applies this method to a field theory, but I have several doubts about how this transition is done.
A first-order Lagrangian for a mechanical system can be written as
\begin{equation}\tag{2.1} L=A_\alpha(\xi)\dot{\xi}^\alpha+L_v(\xi), \end{equation}
whose Euler-Lagrange equations are
\begin{equation}\tag{2.2} F_{\alpha\beta}\dot{\xi}^\beta=-\frac{\partial L_v}{\partial\xi^\alpha}, \end{equation}
with
\begin{equation}\tag{2.3} F_{\alpha\beta}=\frac{\partial A_\beta}{\partial\xi^\alpha}-\frac{\partial A_\alpha}{\partial\xi^\beta} \end{equation}
being the symplectic two-form. So far so good.
Later, on section III.B, he says that the traduction of expresion (2.3) to field theory is
\begin{equation}\tag{3.41} F_{\alpha\beta}=\frac{\delta A_\beta}{\delta\xi^\alpha}-\frac{\delta A_\alpha}{\delta\xi^\beta}, \end{equation}
but I can't see how this araises from Euler-Lagrange equations, i.e.
\begin{equation}\tag{EL} \frac{\delta S}{\delta\phi^\alpha}=\frac{\partial\mathcal{L}}{\partial\phi^\alpha}-\partial_\mu\Bigg(\frac{\partial\mathcal{L}}{\partial({\partial_\mu\phi^\alpha})}\Bigg)=0, \end{equation}
with
\begin{equation} S[\phi]=\int d^4{x} \mathcal{L}(\phi,\partial\phi). \end{equation}
On a field theory, I suppose that a first-orden lagrangian density would be written as
\begin{equation}\tag{LD} \mathcal{L}=A_\alpha(\phi,\nabla\phi)\dot{\phi}^\alpha+\mathcal{L}_v(\phi,\nabla\phi), \end{equation}
where $\phi^\alpha$ are the fields and $\nabla\phi$ indicates dependence in spatial derivatives. However, if I insert that lagrangian density into (EL) I can't get an expression equivalent to (2.2). Moreover, the "$A$"s in expresion (3.41) must be functionals. Which is the relation between those "$A$"s and the ones in (LD)?