Radiation of relativistic ‘harmonic’ oscillator

Abstract

Dynamics and radiation of a relativistic charged particle moving in a linear restoring force field is studied. Solutions to the equations of motion are presented in a compact form in terms of Jacobi elliptic functions. The intensity of radiation, its angular distribution and radiation spectrum of a relativistic oscillator are investigated. It is shown that, unlike a non-relativistic oscillator, the maximum intensity is emitted not at the turning points, but at some intermediate points of the particle's trajectory. The emission spectrum consists of lines at frequencies that are multiples of the oscillation frequency of the oscillator. As the oscillator energy increases, the number of harmonics in the spectrum increases. The maximum in the spectrum of highly relativistic oscillator occurs on high numbers of harmonics.

Non-harmonic response of relativistic particle to driving harmonic force

“…We assume that at the initial time t = 0 v x = 0, where ⃗ v is the particle velocity. In this paper, we neglect the radiation damping force [10], which causes the weak damping of the oscillations over time. It allows us to apply the relativistic form of Newton's second law.…”