On the topology of steady-state solutions of hydrodynamic and vortex consequences of the Vlasov equation and the Hamilton-Jacobi method

Abstract

INTRODUCTIONThe Vlasov–Poisson and Vlasov–Maxwell equations were successfully applied to describe plasma in[1–3]. These equations were reduced to MHDtypeequations by using a hydrodyn amic substitution in [4–8].For the equations of an ideal incompressible fluid,Arnold proved a theorem on the structure of steadystate solutions based on the presence of two commuting vector fields [9]. Kozlov generalized this construction to a contractible fluid [10]. In this paper, we investigate the possibility of such constructions for the Vlasov–Poisson and Vlasov–Maxwell equations.1. THE VLASOV–POISSON EQUATIONSThese equations describe the motion of electrons,electrostatic plasma, or selfgravitating objects, suchas galaxies or galactic clusters. Consider the followingslightly more general case of the Vlasov–Poisson system for gravitational electrostatics [11]:(1)Here, the α denote types of particles,