2 - Lagrangian Coordinates

Abstract

This chapter is devoted to the study of the construction scheme for the theory of Vlasov's kinetic equations. Analytical study of Vlasov equations requires special approaches. The classic one uses Lagrangian coordinates and N-body problem. This chapter discusses different approaches such as hydrodynamics and cosmology to outline additional relations between Vlasov-type equations and physical processes. Some ideas, such as particle method and Hamiltonian dynamics, can be used in different ways, but initially they lead us to the discovery of Vlasov–Maxwell equation. The Vlasov equation accepts the substitution of the form and contains within itself a description of the motion of N bodies for arbitrary number N. Thus, it proves the equation to be fundamental. The Vlasov equation accepts the substitution of the form providing equations for continuum of bodies. If q = (X(0), V(0)) the initial coordinates, then q is called Lagrange coordinates and equation is interpreted as a transformation of it.