Covariant formulation of the Liouville and the Vlasov equations
  The fundamental concept of phase space for particles moving in the four-dimensional spacetime is analyzed. Particle distribution density is defined as differential form, which degree may be different in various cases. It should be emphasized that this approach does not include the concepts of phase volume and distribution function. The Liouville and the Vlasov equations are written in tensor form. The presented approach is valid in both non-relativistic and relativistic cases. It allows using arbitrary systems of coordinates for description of the particle distribution. In some cases, making use of special coordinates gives possibility to construct analytical solutions. Besides, such 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 mass distributions in curved spacetime. 
O.I. Drivotin
physics.class-ph