Canonical formalism of action‐at‐a‐distance electrodynamics and many‐particle potential among charged particles

The second post‐Coulombian Lagrangian of Wheeler–Feynman electrodynamics for a many‐particle system is treated according to a canonical formalism of a singular Lagrangian with higher derivatives. The canonical equations are given in terms of a reduced Hamiltonian with Dirac brackets, but they are transformed to be expressed in terms of ordinary Poisson brackets by redefinition of canonical variables. The reduced Hamiltonian includes a characteristic form of three‐particle and four‐particle potentials. Finally a direct pathway to the reduced Hamiltonian is presented via first‐order formalism of the Maxwell theory with charged particles.