Linearized Variational Analysis of Single-Species, One-Dimensional Vlasov Plasmas

Analysis of linearized Vlasov plasmas by means of Hamilton's variational principle is illustrated by application to the initial-value problem for a neutralized, single-species plasma in one-spatial dimension with periodic boundary conditions. For any degree of approximation and for arbitrary equilibrium velocity distributions, the problem is reduced to the solution of a system of ordinary differential equations in time with constant coefficients and time-dependent driving terms; an exact particular solution has been found. The equations are discussed, including the relation of the equations to Eulerian treatments of the problem, and the use of the equations to obtain numerical results. Numerical examples have been worked out for a Maxwellian equilibrium distribution function, equilibrium distribution functions that are sums of two Maxwellian distributions, and for nonanalytic equilibrium distribution functions that are represented by cubic spline functions. Comparisons are made between numerical results from the variational method and precise solutions of the exact plasma dispersion relation.