The Vlasov equation for systems with velocity dependent interactions

This work contains a discussion of the Vlasov approximation for the system of particles interacting not only via the pair-wise central potential, but also via the velocity dependent interactions (V.D.I). The VDI are assumed to be similar in nature to those encountered in the classical plasma when the higher order in (νc) corrections to the Coulomb forces are taken into account. Using the Klimontovich formulation of the many-body dynamics, we derive the general equation of motion for the exact one-particle distribution function N(r, v, t and we discuss the Vlasov approximation to this equation in some detail. The linearization of the Vlasov equation and the derivation of the dispersion relations for longitudinal and transverse waves are also presented while the detailed analysis of the solutions and of the stability problem is postponed to the following publication.