Post-Newtonian Dynamics of Radiating Charges: Canonical Formulation and Binary Inspiral Laws

We revisit an explicit electromagnetic analogue of the Post Newtonian Hamiltonian framework widely used in gravitational wave physics. Starting from the Lorentz Dirac equation, we implement the Landau-Lifshitz order reduction to cast the $1.5$PN radiation reaction force in terms of a double sum in canonical variables and incorporate this into the well known $1$PN Darwin Hamiltonian system. The resulting phase space is strictly conservative when dissipation is switched off, while in presence of dissipation, it exhibits monotonic energy loss during the inspiral, accompanied by orbit circularization and eccentric bursts in the evolution of the Darwin Hamiltonian. Using this phase space framework we compute the circular and eccentric inspiral laws, including $1$PN conservative corrections. Extending to charged compact binaries in Einstein-Maxwell theory, we combine the $2$PN ADM-type conservative Hamiltonian with leading $1.5$PN dipole dissipation and gravitational quadrupole flux, obtaining gauge-invariant energy-frequency relations, closed-form circular inspiral laws, and a dipole-quadrupole crossover scale that separates electromagnetic and gravitational flux dominated inspirals.