Recalculation of initial conditions when the trajectory of a charged particle crosses the boundary with а jump of electric and magnetic field. I. Electric field

When the trajectories of charged particles in electric fields and/or magnetic fields are simulated, sometimes the case occurs when the trajectory crosses the line with a jump of the electric or magnetic field. It is necessary to recalculate carefully the coordinates and the velocities of the charged particle not to produce strange numerical artifacts like the violation of the conservation of energy. This paper introduces and proves mathematically strictly the principle of refraction - namely, only the normal velocity component should be corrected in such a way that the total energy remains constant while all other velocity components as well as the coordinates of the charged particle just keep their original values.