On Liouville and Vlasov equations, and its hydrodynamic consequences

We describe the derivation of Vlasov–Maxwell equation from classical Lagrangian, and a similar derivation of the Vlasov–Poisson–Poisson charged gravitating particles. We derive electromagnetic hydrodynamic equations and present them to the Godunov's double divergence form. For them we get generalized Lagrange identity and compare it. Analyzes the steady-state solutions of the Vlasov–Poisson–Poisson equation: their types is changing at a certain critical mass having a clear physical meaning. The consequence is the different behavior of particles - recession or collapse trajectories. We investigate topology of solutions of those hydrodynamic equations due to Arnold–Kozlov commuting vector fields, and connection to Hamilton–Jacobi method.