  In many astrophysically relevant situations radiation reaction force acting
upon a charge can not be neglected and the question arises about the location
and stability of circular orbits in such regime. Motion of point charge with
radiation reaction in flat spacetime is described by Lorenz-Dirac (LD)
equation, while in curved spacetime -- by DeWitt-Brehme (DWB) equation
containing the Ricci term and the tail term. We show that for the motion of
elementary particles in vacuum metrics the DWB equation can be reduced to the
covariant form of the LD equation which we use here. Generically, the LD
equation is plagued by runaway solutions, so we discuss computational ways to
avoid this problem in constructing numerical solutions. We also use the first
iteration of the covariant LD equation which is the covariant Landau-Lifshitz
equation, comparing results of these two approaches and showing smallness of
the third-order Schott term in the ultrarelativistic case. We calculate the
corresponding energy and angular momentum loss of a particle and study the
damping of charged particle oscillations around an equilibrium radius. We find
that depending on the orientation of the Lorentz force, the oscillating charged
particle either spirals down to the black hole, or stabilizes the circular
orbit by decaying its oscillations. The later case leads to an interesting new
result of shifting of the particle orbit outwards from the black hole. We also
discuss the astrophysical relevance of the presented approach and provide
estimations of the main parameters of the model.
