Symmetries of the kinetic plasma theory
In recent decades the Lie group analysis has been applied to explore many physically interesting nonlinear problems in gas dynamics, plasma physics etc. Furthermore, different extensions of the classical Lie algorithm to the integro-differential systems of equations of kinetic theory were proposed. In particular, this can be done by using three step algorithm which allows us to obtain symmetries of the kinetic equations from the symmetries of an infinite set of partial differential equations for the moments of distribution functions. First, we determine the symmetry of the truncated system of equations for the moments by the usual Lie procedure. Then, we extend the obtained symmetry transforms to the case of an infinite set of the equations. Finally, we restore the symmetries of the integro-differential kinetic theory equations using the generating functional of the above mentioned moments.
In the present work, this indirect algorithm is illustrated on the specific example of the Langmuir waves in the multi-component plasma. Lie symmetries are obtained and it is shown that kinetic symmetry group is larger than hydrodynamic one for the plasmas containing components with equal charge to mass ratio of particles. The same algorithm is applied to the more complicated problem (two dimensional both in coordinate as well as in velocity space) of the high frequency waves in electron plasma in the constant external magnetic field, i.e. for the upper hybrid waves. Lie symmetries are obtained, together with their extension in the cold electron plasma case. It is emphasized that such symmetry extension allows to obtain exact, but implicit, general solution in Lagrangian variables.