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.