Covariant description of phase space distributions

The concept of phase space for particles moving in the 4-dimensional space time is formulated. Definition of particle distribution density as differential form is given. The degree of the distribution density form may be different in various cases. The Liouville and the Vlasov equations are written in tensor form with use of such tensor operations as the Lie dragging and the Lie derivative. The presented approach is valid in both non-relativistic and relativistic cases. It should be emphasized that this approach does not include the concepts of phase volume and distribution function. The covariant approach allows using arbitrary systems of coordinates for description of the particle distribution. In some cases, making use of special coordinates grants the possibility to construct analytical solutions. Besides, such an approach is convenient for description of degenerate distributions, for example, of the Kapchinsky–Vladimirsky distribution, which is well-known in the theory of charged particle beams. It can be also applied for description of particle distributions in curved space time. Refs 25.