Synchrotron motion with radiation reaction

It is shown that the longitudinal velocity of a charged particle moving in a uniform magnetic field, and obeying Dirac-Lorentz relativistic equation of motion with radiation reaction is constant. Suitable approximate methods, which give fairly accurate results, are used to obtain the expression for velocity and displacement along the transverse section. They describe the motion completely up to a correcting factor $$1 + 0\left\{ {\left( {\frac{{e^3 B}}{{m^3 c^4 }}} \right)^2 } \right\}; \frac{{e^3 B}}{{m^3 c^4 }} \simeq 10^{ - 16} B$$ for electrons,B inG.