Write $D=\partial_\mu A^\mu$. The additional <gauge fixing> variation is $\delta(-D^2/2)=-D\partial_\mu\delta A^\mu$. <Integration by parts> gives the <Euler-Lagrange field equation> $\partial_\mu F^{\mu\nu}+\partial^\nu D=0$, hence $\Box A^\nu=0$. This equation follows from the gauge-fixed density without imposing $D=0$ as a separate operator identity.

Direct differentiation of the density as printed gives the <canonical momentum> components conjugate to the lower-index fields:
$$
\boxed{\pi^\nu=\frac{\partial\mathcal L}{\partial\dot A_\nu}=-F^{0\nu}-\eta^{0\nu}D,\qquad \pi^0=-D,\quad\pi^i=\dot A_i-\partial_iA_0.}
$$
There is a boundary-term convention to reconcile with part (c). The <Feynman-gauge Maxwell kinetic density after a boundary-term subtraction> is
$$
\mathcal L'=-\frac12\partial_\mu A_\nu\partial^\mu A^\nu,
\qquad
\mathcal L=\mathcal L'+\partial_\mu K^\mu,
\qquad
K^\mu=\frac12\bigl(A_\nu\partial^\nu A^\mu-A^\mu D\bigr).
$$
It has the same <Euler-Lagrange field equations>, but its <canonical momentum> components are $\pi'^\nu=-\dot A^\nu$. The mode expansion supplied in part (c) is the expansion of these latter momenta. Thus it is valid after the stated boundary-term subtraction; it is not the direct derivative of the original density. For example, a time-independent spatially varying $A_0$ with $A_i=0$ gives $\pi^i=-\partial_iA_0$ but $\pi'^i=0$.

The <boundary-term shift of canonical field momenta> is a <canonical transformation>. If spatial boundary terms vanish, $B=\int d^3x\,K^0=\int d^3x\,A_0\partial_iA_i$, and $\pi^\nu=\pi'^\nu+\delta B/\delta A_\nu$. This explains why the two conventions have the same dynamics while their <momentum> formulas differ.

Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 301 / 1 / b / Solution

If the spatially integrated <Lagrangian> changes by $dB[\Phi]/dt$, where $B$ depends on fields and their spatial derivatives but not velocities, the <canonical momentum> components change by $\pi_a\mapsto\pi_a+\delta B/\delta\Phi_a$. This follows by varying $dB/dt$ and using spatial <integration by parts>. The canonical one-form changes by the exact variation $\delta B$, so the <symplectic form> and <canonical commutation relations> are preserved. For the two <Feynman gauge> Maxwell densities, $B=\int A_0\partial_iA_i\,d^3x$ gives shifts $\Delta\pi^0=\partial_iA_i$ and $\Delta\pi^i=-\partial_iA_0$. Boundary flux must vanish for this spatial functional calculation.

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 Incoming links: Boundary-term shift of canonical field momenta