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Vlasov-type and Liouville-type equations, their microscopic, energetic and hydrodynamical consequences

We give a derivation of the Vlasov–Maxwell and Vlasov– Poisson–Poisson equations from the Lagrangians of classical electrodynamics. The equations of electromagnetic hydrodynamics (EMHD) and electrostatics with gravitation are derived from them by means of a `hydrodynamical' substitution. We obtain and compare the Lagrange identities for various types of Vlasov equations and EMHD equations. We discuss the advantages of writing the EMHD equations in Godunov's double divergence form. We analyze stationary solutions of the Vlasov– Poisson– Poisson equation, which give rise to non-linear elliptic equations with various properties and various kinds of behaviour of the trajectories of particles as the mass passes through a critical value. We show that the classical equations can be derived from the Liouville equation by the Hamilton–Jacobi method and give an analogue of this procedure for the Vlasov equation as well as in the non-Hamiltonian case.