It is generally accepted that the dynamics of relativistic particles in the lab frame can be described by taking into account the relativistic dependence of the particles momenta on the velocity, with no reference to Lorentz transformations. The electrodynamics problem can then be treated within a "single inertial frame" description. To evaluate radiation fields from moving charged particles we need their velocities and positions as a function of the lab frame time t. The relativistic motion of a particle in the lab frame is described by Newton's second law corrected for the relativistic dependence of the particle momentum on the velocity. In all standard derivations the trajectories in the source part of the usual Maxwell's equations are identified with the trajectories $\vec{x}(t)$ calculated by using the "corrected" Newton's second law. This way of coupling fields and particles is considered correct. We argue that this procedure needs to be changed by demonstrating a counterintuitive: the results of conventional theory of radiation by relativistically moving charges are not consistent with the principle of relativity. The trajectory of a particle in the lab frame consistent with the usual Maxwell's equations, is found by solving the dynamics equation in manifestly covariant form, with the proper time $\tau$ used to parameterize the particle world-line in space-time. We find a difference between the "true" particle trajectory $\vec{x}(t)$ calculated or measured in the conventional way, and the covariant particle trajectory $\vec{x}_{cov}(t)$ calculated by projecting the world-line to the lab frame and using t to parameterize the trajectory curve. The difference is due to a choice of convention, but only $\vec{x}_{cov}(t)$ is consistent with the usual Maxwell's equations: therefore, a correction of the conventional synchrotron-cyclotron radiation theory is required.