Motion of charged particles in slowly varying fields to the first order of approximation

Following Bogolyubov's and Zubarev's perturbation method, we analyze the motion of a charged particle to the first order of approximation. The general case of motion in a, time-dependent nonhomogeneous magnetic field in the presence of an electric field and currents, as well as simpler cases, is considered. Correction terms of the average kinetic energy, the acceleration, and the position of mirror points, as well as a formula for the first adiabatic invariant to 1/Ω order, were obtained. The guiding center's mirror points differ, in all cases, from the points of magnetic reflection of the particle, which contain an oscillatory term and depend on the average, phase angle. Only for the case of a static, nonhomogeneous field, in the absence of an electric field and currents, the magnetic moment is conserved and all the first-order correction terms vanish.