Extension of Faddeev-Jackiw first-order Lagrangian formalism to fields 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)?
 A: OP is essentially asking the following.

How does the point-mechanical Faddeev-Jackiw formulas from chapter II generalizes to the field-theoretic formulas in section III.B?

Well, let's see. The field-theoretic first-order Lagrangian is
$$ L~=~\int\!d^3{\bf x}~A_{\alpha}({\bf x}) \dot{\xi}^{\alpha}({\bf x})~-~H.\tag{2.1'}$$
An infinitesimal variation reads
$$\begin{align} \delta L
~\sim~& \int\!d^3{\bf x}\left( \delta A_{\alpha}({\bf x}) \dot{\xi}^{\alpha}({\bf x})
 -\frac{dA_{\alpha}({\bf x})}{dt}  \delta\xi^{\alpha}({\bf x})\right)
~-~\delta H \cr
~=~&\int\!d^3{\bf x} \int\!d^3{\bf x}^{\prime}  \left( \delta\xi^{\alpha}({\bf x}) 
\frac{\delta A_{\beta}({\bf x}^{\prime})}{\delta \xi^{\alpha}({\bf x})}  \dot{\xi}^{\beta}({\bf x}^{\prime}) 
~-~ \dot{\xi}^{\beta}({\bf x}^{\prime}) \frac{\delta A_{\alpha}({\bf x})}{\delta \xi^{\beta}({\bf x}^{\prime})}  \delta\xi^{\alpha}({\bf x}) \right) \cr
&~-~ \delta  H\cr
~=~&\int\!d^3{\bf x} \int\!d^3{\bf x}^{\prime} \delta\xi^{\alpha}({\bf x})  F_{\alpha\beta}({\bf x},{\bf x}^{\prime})\dot{\xi}^{\beta}({\bf x}^{\prime}) 
~-~ \int\!d^3{\bf x}~ \delta \xi^{\alpha}({\bf x})  \frac{\delta  H}{\delta \xi^{\alpha}({\bf x})}.
\end{align}$$
Here the $\sim$ symbol means equality modulo total time derivative terms. Also we have defined the components of symplectic 2-form
$$
F_{\alpha\beta}({\bf x},{\bf x}^{\prime})~=~\frac{\delta A_{\beta}({\bf x}^{\prime})}{\delta\xi^{\alpha}({\bf x})}-\frac{\delta A_{\alpha}({\bf x})}{\delta\xi^{\beta}({\bf x}^{\prime})}.\tag{2.3'/3.41'}
$$
(Judging from eq. (3.42) it becomes clear that eq. (3.41) is bi-local.)
The Euler-Lagrange equations are Hamilton's equations
$$ \int\!d^3{\bf x}^{\prime}   F_{\alpha\beta}({\bf x},{\bf x}^{\prime})\dot{\xi}^{\beta}({\bf x}^{\prime}) 
~=~   \frac{\delta  H}{\delta \xi^{\alpha}({\bf x})}.\tag{2.2'}
$$
References:

*

*D.J. Toms, Faddeev-Jackiw quantization and the path integral, arXiv:1508.07432.

