Dissipative Potentials and the Motion of a Classical Charge. II

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Abstract
In an earlier paper by the author, examples of the motion of a point charge were found to be consistent with the hypothesis of Abraham that the mass of an electron (or positron) is entirely electromagnetic. Further consequences of this hypothesis are developed. It is shown that the conservation laws of the electromagnetic field and Maxwell's equations require that the total Lorentz force (including the self-force) on the charge should vanish. This result can be expressed as a Lagrangian equation of motion. The canonical four momentum of the charge is the product of the magnitude of the charge by the four potential of the field at the position of the charge. When the dissipative form of the potential for an unconfined point charge is used, the integro-differential equation of motion of the earlier paper is obtained for a particle with zero "bare" mass. A mechanical momentum and mass are defined; these are associated with the singular part of the Green's function for the D'Alembert equation. The rate of change of this mechanical momentum is equal to the sum of the external force, the radiation damping force (with the correct sign obtained by the use of the retarded fields), and the gradient at the position of the charge of its Coulombic self-potential energy. For a particle assumed to follow a continuous trajectory, the integrals in the integro-differential equation of motion are evaluated by a procedure in agreement with, but much simpler than, that of Dirac. The result is the unrenormalized equation of Dirac for a particle whose mass is the divergent Coulombic self-energy. The effective momentum and mass in this equation are reduced to half of the mechanical momentum and mass by the force term arising from the gradient of the Coulombic self-potential energy.

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