Acceleration and Particle Field Interactions of Cosmic Rays I: Formalism

The acceleration of ultra high energy cosmic rays is conjectured to occur through various interactions with the electromagnetic fields in different astrophysical objects, like magnetic matter clumps, besides the well-known shock and stochastic Fermi mechanisms. It is apparent that the latter are not depending on the particle's charge, quantitatively. Based on this model, a considerable portion of the dynamics, that derives a charged particle parallel to a magnetic field $\mathbf{B}$ and under the influence of a force $\mathbf{F}$, is assumed to be composed of an acceleration by a non-magnetic force $\mathbf{F}_{\parallel}$ and a gyromotion along $\mathbf{B}$ direction, plus drifts in the direction of $\mathbf{F}_{\perp}$. The model and its formalism are introduced. Various examples for drift motions and accelerating forces are suggested. The formalism is given in a non-relativistic version. Obviously, the translation into the relativistic version is standard. In a forthcoming work, a quantitative estimation of the energy gained by charged cosmic rays in various astrophysical objects will be evaluated.