An electric charge moving in an external electromagnetic field can emit electromagnetic radiation. Thus, apart from the Lorenz force, one needs to consider the "recoil" caused by the emitted radiation. To put it in another way, the charge is affected by its own field. However, this is not so straightforward to take account of, as the field of a point charge at its location is singular.
The usual way to compute the motion of a charged particle in an external electromagentic field is to rely on the so-called Lorenz-Dirac equation. This equation has a number of known problems. In particular, it is a third order equation of motion, thus apart from initial position and velocity one needs to "prescribe" an initial acceleration to uniquely determine a solution. Also, it admits physically unacceptable runaway solutions.
Often these problems are treated in an ad-hoc manner. We take the viewpoint that the physically acceptable solutions of the Lorentz-Dirac equation are actually determined by some second order equation of motion. Then, postulating certain natural properties, in some cases we are actually able to determine a concrete form for this equation.