Advanced Theory of Gyrating Particles

Let a magnetic field B and an electric field E be given as functions of position r and time t. We are interested in the non-relativistic motion of a charged ''test'' particle in the familiar ''guiding-centre'' approximation. This applies when the motion of the particle may be treated approximately as a gyration (uniform circular motion) orthogonal to B, of small radius but finite velocity (hence large frequency), superimposed on the finite velocity motion of a so-called guiding-centre. The electromagnetic fields at the position of the particle change only a little during one gyration period