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We develop some ideas about gauge symmetry in the context of Maxwell's theoryof electromagnetism in the Hamiltonian formalism. One great benefit of thisformalism is that it pairs momentum and configurational degrees of freedom, sothat a decomposition of one side into subsets can be translated into adecomposition of the other. In the case of electromagnetism, this enables us topair degrees of freedom of the electric field with degrees of freedom of thevector potential. Another benefit is that the formalism algorithmicallyidentifies subsets of the equations of motion that represent time-dependentsymmetries. For electromagnetism, these two benefits allow us to definegauge-fixing in parallel to special decompositions of the electric field. Morespecifically, we apply the Helmholtz decomposition theorem to split theelectric field into its Coulombic and radiative parts, and show how this givesa special role to the Coulomb gauge (i.e. div$({\bf A}) = 0$). We relate thisargument to Maudlin's (2018) discussion, which advocated the Coulomb gauge.

How to Choose a Gauge? The case of Hamiltonian Electromagnetism