Following particle trajectories in the intense electromagnetic field of a neutron star is prohibited by the large ratio between the cyclotron frequency~$\omega_{\rm B}$ and the stellar rotation frequency~$\Omega$. No fully kinetic simulations on a macroscopic scale and with realistic field strengths have been performed so far due to the huge computational cost implied by this enormous scale separation. In this paper, we derive new expressions for the particle velocity subject to strong radiation reaction and intended to be more accurate than the current state of the art expression in the radiation reaction limit regime, the so called Aristotelian regime. We short cut the time scale hierarchy by solving the particle equation of motion in the radiation reaction regime where the Lorentz force is always and immediately balanced by the radiative drag including a friction not necessarily opposite to the velocity vector, as derived in the Landau-Lifshitz approximation. Starting from the reduced Landau-Lifshitz equation we find expressions for the velocity depending only on the local electromagnetic field configuration and on a new parameter controlling the strength of the radiative damping, related to the field strength. As an example, we impose a constant Lorentz factor~$\gamma$ during the particle motion. We found that for ultra-relativistic velocities satisfying $\gamma \gtrsim 10$, the difference between strong radiation reaction and the radiation reaction limit becomes negligible. The new velocity expressions produce results similar in accuracy to the radiation reaction limit approximation. We therefore do not expect this new method to improve the accuracy of neutron star magnetosphere simulations. The radiation reaction limit is a simple but accurate, robust and efficient way to follow ultra-relativistic particles in a strong electromagnetic field.