Degenerate solutions of electron-beam equations

In this paper, exact solutions are constructed for stationary election beams that are degenerate in the Cartesian (x,y,z), axisymmetric (r,θ,z), and spiral (in the planes y=const (u,y,v)) coordinate systems. The degeneracy is determined by the fact that at least two coordinates in such a solution are cyclic or are integrals of motion. Mainly, rotational beams are considered. Invariant solutions for beams in which the presence of vorticity resulted in a linear dependence of the electric-field potential ? on the above coordinates were considered in 1], In degenerate solutions, the presence of vorticity results in a quadratic or more complex dependence of the potential on the coordinates that are integrals of motion. In 2] and in a number of papers referred to in 2], the degenerate states of irrotational beams are described. The known degenerate solutions for rotational beams apply to an axisymmetric one-dimensional (r) beam with an azimuthal velocity component 3] and to relativistic conical flow 1]. The equations used below follow from the system of electron hydrodynamic equations for a stationary relativistic beam $$\begin{array}{*{20}c} {\sum\limits_{\beta = 1}^3 {\frac{\partial }{{\partial q^\beta }}\left {\sqrt \gamma g^{\beta \beta } g^{\alpha \alpha } \left( {\frac{{\partial A_\alpha }}{{\partial q^\beta }} - \frac{{\partial A_\beta }}{{\partial q^\alpha }}} \right)} \right]} = 4\pi \rho \sqrt \gamma g^{\alpha \alpha } u_\alpha ,} \\ {\sum\limits_{\beta = 1}^3 {\frac{\partial }{{\partial q^\beta }}\left( {\sqrt \gamma g^{\beta \beta } \frac{{\partial \varphi }}{{\partial q^\beta }}} \right)} = 4\pi \rho \sqrt {\gamma u} ,\sum\limits_{\beta = 1}^3 {g^{\beta \beta } u_\beta ^2 + 1 = u^2 } } \\ \begin{gathered} \frac{\eta }{c}u\frac{{\partial \mathcal{E}}}{{\partial q^\alpha }} = \sum\limits_{\beta = 1}^3 {g^{\beta \beta } u_\beta } \left( {\frac{{\partial p_\beta }}{{\partial q^\alpha }} - \frac{{\partial p_\alpha }}{{\partial q^\beta }}} \right), \hfill \\ \begin{array}{*{20}c} {\sum\limits_{\beta = 1}^3 {\frac{\partial }{{\partial q^\beta }}(\sqrt \gamma g^{\beta \beta } \rho u_\beta ) = 0,u \equiv \frac{\eta }{{c^2 }}(\varphi + \mathcal{E}) + 1,} } \\ {cu_\alpha \equiv \frac{\eta }{c}A_\alpha + p_\alpha ,\alpha ,\beta = 1,2,3,\gamma \equiv g_{11} g_{22} g_{33} } \\ \end{array} \hfill \\ \end{gathered} \\ \end{array} $$ where q^β denotes orthogonal coordinates with the metric tensor g^ββ (β=1,2,3); A_α is the magnetic potential; A_α = (u_α/u)c is the electron velocity; ρ is the scalar space-charge density (ρ > 0); is the energy in eV; p_α is the generalized momentum of an electron per unit mass; η is the electron charge-mass ratio.