In the seemingly standard treatment of classical electromagnetism (cf. Qmechanic's answer here) we decide to work entirely within the framework of classical field theory. In particular, we define our dynamic variables as the electromagnetic potential $\mathcal{A}$ and associated matter fields $\psi$ and then define a Lagrangian $\mathcal{L}(\mathcal{A}, \psi)$. Concretely, we might study a theory of the electron in which we let $\psi$ be a Grassmann-valued Dirac field. But, this sort of treatment seems to stem from attempting to classicalize quantum field theory.

Is there a way to treat classical electromagnetism in which particles are treated using Lagrangian mechanics and the electromagnetic field is treated using Lagrangian field theory? This might entail, for instance, writing down some particle-field Lagrangian.

In the seemingly standard treatment of classical electromagnetism (cf. Qmechanic's answer here) we decide to work entirely within the framework of classical field theory. In particular, we define our dynamic variables as the electromagnetic potential $\mathcal{A}$ and associated matter fields $\psi$ and then define a Lagrangian $\mathcal{L}(\mathcal{A}, \psi)$. Concretely, we might study a theory of the electron in which we let $\psi$ be a Grassmann-valued Dirac field. But, this sort of treatment seems to stem from attempting to classicalize quantum field theory.

Is there a way to treat classical electromagnetism in which particles are treated using Lagrangian mechanics and the electromagnetic field is treated using Lagrangian field theory in a unified manner? This might entail, for instance, writing down some particle-field Lagrangian.

(Of course, textbook electromagnetism uses a particle-field formalism, but such formalism seems put together in an ad-hoc manner.)