A variational principle and an energy theorem for small-amplitude disturbances of electron beams and of electron-ion plasmas

Abstract

Small-amplitude disturbances of electron beam and electromagnetic field configurations may be conveniently represented by the electromagnetic potentials of the field perturbation and the displacement (or “polarization”) vector representing the perturbation of the electron assembly. With this choice of dynamical variables, it is possible to establish a Lagrangian function for small-amplitude disturbances, in which both field and particles are described by “field-like” variables. This derivation is carried through in relativistically covariant form. It is shown that, by a variant of a formula due to Schrödinger, we may establish an acceptable formula for the energy-momentum tensor of the field-particle assembly. The more important formulas are given also in nonrelativistic three-dimensional form. The differential equation satisfied by the energy-momentum tensor represents a generalization of the so-called “power theorem” of electron-tube theory. It is anticipated that this variational principle, which may be readily extended to include a number of beams, or a distribution of electron velocities, will provide a convenient basis for certain electron-tube analyses, and for the investigation of certain stability problems in plasma physics.