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_{alphabeta}dot{xi}^beta=-frac{partial L_v}{partialxi^alpha},
end{equation}

with

begin{equation}tag{2.3}
F_{alphabeta}=frac{partial A_beta}{partialxi^alpha}-frac{partial A_alpha}{partialxi^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_{alphabeta}=frac{delta A_beta}{deltaxi^alpha}-frac{delta A_alpha}{deltaxi^beta},
end{equation}

but I can’t see how this araises from Euler-Lagrange equations, i.e.

begin{equation}tag{EL}
frac{delta S}{deltaphi^alpha}=frac{partialmathcal{L}}{partialphi^alpha}-partial_muBigg(frac{partialmathcal{L}}{partial({partial_muphi^alpha})}Bigg)=0,
end{equation}

with

begin{equation}
S[phi]=int d^4{x} mathcal{L}(phi,partialphi).
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,nablaphi)dot{phi}^alpha+mathcal{L}_v(phi,nablaphi),
end{equation}

where $phi^alpha$ are the fields and $nablaphi$ 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)?