= Boundary-term shift of canonical field momenta
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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