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

We construct an explicit electromagnetic analogue of the post-Newtonian Hamiltonian framework widely used in gravitational-wave physics. Starting from the Lorentz--Dirac equation and implementing Landau--Lifshitz order reduction, we derive the near-zone 1.5PN dipole radiation-reaction force and combine it with the Darwin Hamiltonian through 1PN order to obtain a closed canonical N-body phase-space system. The equations are directly implementable, strictly conservative when dissipation is switched off, and exhibit monotonic energy loss, secular inspiral, and circularization when radiation reaction is included, accompanied by eccentric bursts in the evolution of the Darwin Hamiltonian. For binaries we derive analytic circular and eccentric inspiral laws, including 1PN conservative corrections to the dipole-driven inspiral and merger time for the circular orbit. Extending to charged compact binaries in Einstein-Maxwell theory, we combine the 2PN ADM-type conservative Hamiltonian with leading 1.5PN 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.