| A detailed analysis of the noncanonical structure of the linearized Maxwell-Vlasov equations is presented. The full Maxwell-Vlasov bracket is linearized about a stable, homogeneous and isotropic equilibrium. Velocity space moments are taken leading to a natural decomposition of the system into longitudinal and transverse parts. This bracket together with the linearized energy is shown to give the usual linearized moment equations. A family of integral transforms whose kernels are closely related to singular eigenfunctions are introduced. It is shown that by means of these transformations, both the bracket and energy can be brought into diagonal form. The diagonalizing transformation is essentially a transformation to linear action-angle variables for this infinite-dimensional Hamiltonian system. The resulting energy expression, which depends on the Fourier transform of the electric field, has physical meaning as the energy of the perturbations and in general is not equal to the usual expression for the wave energy in a dielectric. Equilibria that support discrete modes are also studied. It is shown that the eigenfunctions corresponding to the discrete modes enter as a natural result of regularizing the (now singular) inverse transform. It is seen that in the case of neutral modes, the transformed variables must be interpreted as generalized functions. Lastly, quadrature rules for Cauchy integrals are discussed and an efficient, high accuracy algorithm for computing Hilbert transforms is developed. This algorithm is used to evaluate the exact solution of the longitudinal equations for different initial conditions. |
