Hamiltonian formulation for the classical EM radiation-reaction problem: Application to the kinetic theory for relativistic collisionless plasmas

Abstract

A notorious difficulty in the covariant dynamics of classical charged particles subject to non-local electromagnetic (EM) interactions arising in the EM radiation-reaction (RR) phenomena is due to the definition of the related non-local Lagrangian and Hamiltonian systems. As a basic consequence, the lack of a standard Lagrangian/Hamiltonian formulation in the customary asymptotic approximation for the RR equation may inhibit the construction of consistent kinetic and fluid theories. In this paper the issue is investigated in the framework of Special Relativity. It is shown that, for finite-size spherically-symmetric classical charged particles, non-perturbative Lagrangian and Hamiltonian formulations in standard form can be obtained, which describe particle dynamics in the presence of the exact EM RR self-force. As a remarkable consequence, based on axiomatic formulation of classical statistical mechanics, the covariant kinetic theory for systems of charged particles subject to the EM RR self-force is formulated in Hamiltonian form. A fundamental feature is that the non-local effects enter the kinetic equation only through the retarded particle 4-position. This permits, in turn, the construction of the related fluid equations, in which the non-local contributions carried by the RR effects are explicitly displayed. In particular, it is shown that the moment equations obtained in this way do not contain higher-order moments, allowing as a consequence the adoption of standard closure conditions. A remarkable aspect of the theory is related to the short delay-time asymptotic expansions. Here it is shown that two possible expansions are permitted. Both can be implemented for the single-particle dynamics as well as for the corresponding kinetic and fluid treatments. In the last case, they are performed a posteriori, namely on the relevant moment equations obtained after integration of the kinetic equation over the velocity space. Comparisons with the literature are pointed out.