Post-newtonian dynamics of radiating charges: canonical formulation and binary inspiral laws

Abstract 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 1.5PN radiation reaction force in terms of a double sum in canonical variables and incorporate this into the well known 1PN 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 1PN conservative corrections. Extending to charged compact binaries in Einstein-Maxwell theory, we combine the known 2PN ADM-type conservative Hamiltonian with leading 1.5PN dipole dissipation and 2.5PN gravitational quadrupole flux, obtaining gauge-invariant energy-frequency relations, closed-form inspiral laws, and a dipole-quadrupole crossover scale that separates electromagnetic and gravitational flux dominated inspirals.