Chapter Six - Geometrized Theory

Abstract

It is shown that the joint use of the beam equations in the coordinate system associated with the flow's geometry and the Euclidean conditions for this system's metrics allows formulation of a complete system of equations that include the motion equations, Maxwell equations, and three of the six Lame identities containing the second longitudinal derivatives of the metrics tensor elements. For axisymmetric flows, the system of geometrized equations includes a relation on the stream tube, which does not contain the transversal derivatives, and a system of evolutionary equations, which allows a transition from the basic surface to a neighboring stream tube. This offers an opportunity for synthesis of non-paraxial solid or tubular beams using the narrow bands method or the theory of higher approximations. Investigation of the characteristics of the geometrized equations reveals the existence of real and complex solutions of the defining equation, which makes it impossible to use the known stable numerical algorithms to solve the Cauchy problem with access to the complex space. These algorithms are valid only in the case of purely imaginary characteristics. Combined models are constructed to provide a detailed description of the emitter's vicinity and the calculation of the trajectories in the first approximation. As a result, a relationship is established between the potential distribution on the basic surface and the derivatives of the emitter curvatures and the emission current density. This offers an opportunity to control the emitter shape and the current density on the emitter. A geometrized theory of the narrow spatial relativistic beams and higher approximations for the electrostatic flows are considered. It is shown that “common sense” considerations reveal their inconsistency when dealing with the geometrized model: Formal operations that have no physical meaning will inevitably lead to some physical consequences. An approach of a priori postulation of the stream tubes adjoins the general geometrized theory. This approach can be realized only for the beams with elliptical cross section, however, it is much simpler than the geometrized theory. Within the framework of this model, a theory of relativistic beams with arbitrary spatial axis in a nonhomogeneous external field is constructed.