  For more than a century the Abraham-Lorentz equation has generally been
regarded as the correct description of the dynamics of a charged particle.
However, there are pathological solutions of the Abraham-Lorentz equation in
which a particle accelerates in advance of the application of a force, the
so-called preacceleration solutions, and solutions in which the particle
spontaneously accelerates even in the absence of an external force, also known
as runaway solutions. Runaways violate conservation of energy while
preacceleration violates causality. In this work, I will focus on one of the
most used alternative equations of motion: the Landau-Lifshitz equation, which
has no pathological solution. However, it is a first-order approximation to the
Abraham-Lorentz equation, raising the question of how an approximation turns
out to be better than the original. Finally, some numerical results for a
variety of external forces are presented to compare both the equations.
