Описана история развития сотрудничества коллектива ИПМ им. М.В. Келдыша РАН с С.К. Годуновым. В процессе этого сотрудничества получено много интересных результатов в теории кинетических уравнений и вычислительной математике. Библ. 20.

The history of the cooperation between the staff of the Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences and S.K. Godunov is described. Numerous novel and interesting results in the theory of kinetic equations and computational mathematics were obtained in the course of this cooperation.

We give a derivation of the Vlasov-Maxwell and VlasovPoisson- 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.

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.