  In accelerator and plasma physics it is accepted that there is no need to
solve the dynamical equations for particles in covariant form, i.e. by using
the coordinate-independent proper time to parameterize particle world-lines in
space-time: to describe dynamics in the laboratory frame, there is no need to
use the laws of relativistic kinematics. It is sufficient to account for the
relativistic dependence of particles momenta on the velocity in the second
Newton's law. Then, the coupling of fields and particles is based on the use of
result from particle dynamics treated according to Newton's laws in terms of
the relativistic three-momentum and on the use of Maxwell's equations in
standard form. Previously, we argued that this is a misconception. Here we
describe in detail how to calculate the coupling between fields and particles
in a correct way and how to develop a new algorithm for a particle tracking
code in agreement with the use of Maxwell's equations in their standard form.
Advanced textbooks on classical electrodynamics correctly tell us that
Maxwell's equations in standard form in the laboratory frame and charged
particles are coupled by introducing particles trajectories as projections of
particles world-lines onto coordinates of the laboratory frame and then by
using the laboratory time to parameterize the trajectory curves. We show a
difference between conventional and covariant particle tracking results in the
laboratory frame. This essential point has never received attention in the
physical community. Only the solution of the dynamical equations in covariant
form gives the correct coupling between field equations in standard form and
particles trajectories in the laboratory frame. Previous theoretical and
simulation results in accelerator and plasma physics should be re-examined in
the light of the pointed difference between conventional and covariant particle
tracking.
