On derivation and classification of Vlasov type equations and equations of magnetohydrodynamics. The Lagrange identity, the Godunov form, and critical mass

V. V. Vedenyapin ; M. A. Negmatov

Contemporary Mathematics. Fundamental Directions, Proceedings of the Sixth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2011). Part 3, Tome 47 (2013), pp. 5-17

Résumé

The derivation of the Vlasov–Maxwell and the Vlasov–Poisson–Poisson equations from Lagrangians of classical electrodynamics is described. The equations of electromagnetohydrodynamics (EMHD) type and electrostatics with gravitation are obtained. We obtain and compare the Lagrange equalities and their generalizations for different types of the Vlasov and EMHD equations. The conveniences of writing the EMHD equations in twice divergent form are discussed. We analyze exact solutions to the Vlasov–Poisson–Poisson equations with the presence of gravitation where we have different types of nonlinear elliptic equations for trajectories of particles with critical mass $m^2=e^2/G$, which has an obvious physical sense, where $G$ denotes the gravitation constant and $e$ is the electron charge. As a consequence we have different behaviors of particles: divergence or collapse of their trajectories.