Derivation and classification of Vlasov-type and magnetohydrodynamics equations: Lagrange identity and Godunov's form

We describe the derivation of the Vlasov–Maxwell equations from the Lagrangian of classical electrodynamics, from which magnetohydrodynamic-type equations are in turn derived. We consider both the relativistic and nonrelativistic cases: with zero temperature as the exact consequence of the Vlasov–Maxwell equations and with nonzero temperature as a zeroth-order approximation of the Maxwell–Chapman–Enskog method. We obtain the Lagrangian identities and their generalizations for these cases and compare them.