Dielectric energy versus plasma energy, and Hamiltonian action-angle variables for the Vlasov equation

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Abstract
Expressions for the energy content of one‐dimensional electrostatic perturbations about homogeneous equilibria are revisited. The well‐known dielectric energy, ED, is compared with the exact plasma free energy expression, δ2F, that is conserved by the Vlasov–Poisson system [Phys. Rev. A 40, 3898 (1989) and Phys. Fluids B 2, 1105 (1990)]. The former is an expression in terms of the perturbed electric field amplitude, while the latter is determined by a generating function, which describes perturbations of the distribution function that respect the important constraint of dynamical accessibility of the system. Thus the comparison requires solving the Vlasov equation for such a perturbation of the distribution function in terms of the electric field. This is done for neutral modes of oscillation that occur for equilibria with stationary inflection points, and it is seen that for these special modes δ2F=ED. In the case of unstable and corresponding damped modes it is seen that δ2F≠ED; in fact δ2F≡0. This failure of the dielectric energy expression persists even for arbitrarily small growth and damping rates since ED is nonzero in this limit, whereas δ2F remains zero. In the case of general perturbations about stable equilibria, the two expressions are not equivalent; the exact energy density is given by an expression proportional to ω‖E(k,ω)‖2‖ε(k,ω)‖2/εI(k,ω), where E(k,ω) is the Fourier transform in space and time of the perturbed electric field (or equivalently the electric field associated with a single Van Kampen mode) and ε(k,ω) is the dielectric function with ω and k real and independent. The connection between the new exact energy expression and the at‐best approximate ED is described. The new expression motivates natural definitions of Hamiltonian action variables and signature. A general linear integral transform (or equivalently a coordinate transformation) is introduced that maps the linear version of the noncanonical Hamiltonian structure, which describes the Vlasov equation, to action‐angle (diagonal) form.