Classical relativistic electron-field dynamics: Hamiltonian approach to radiation reaction

Abstract

Motivated by the renewed interest due to the presently available extreme light sources, the dynamics of a single classical relativistic (spinless) extended electron interacting with a classical electromagnetic field (an incoming radiation and the field radiated by the electron) is revisited. The field is treated in Lorentz gauge, with the Lorentz condition. By assumption, there is a crucial finite cut-off k max on the magnitude of any wavevector contributing to the field (preventing a point electron) and, for a simple formulation, the initial conditions for particle and fields are given in the infinitely remote past. In an infinite three-dimensional vacuum and in an inertial system, Hamilton’s dynamical equations for the particle and the complex field amplitudes acting as canonical variables (a's) yield an exact Lorentz force equation for the former, that includes the incoming radiation and an exact radiation reaction force F RR due to the field radiated by the electron. Uniform motion is obtained as a test of consistency. Based upon numerical computations, some approximations on F RR are given. A covariant formulation is also presented.