Textbook in PDF format

There are three approaches to theoretical consideration of dense beams: (1) numerical integration of the suitably formulated partial differential equations, (2) finding their exact solutions, and (3) constructing an asymptotic theory with a small parameter inherent to a certain class of phenomena. The book analyzes all of these approaches. It discusses the tensor form of the equations that describe the laminar beam, considered as a continuous medium subject to the action of the self-consistent electromagnetic field, formulation of the geometrized form of these equations, and discussion of the single-component flow theory in the light of the geometrized approach. Basing the discussion on the group properties of the beam equations, the book outlines a modern view of finding the exact solutions to these equations. In accordance with the terminology accepted in the mechanics of a continuous medium, one speaks about the exact solution if the original system of partial differential equations describing the dense beam can be reduced to a system of ordinary differential equations. This definition of exact solution is more general than the solution that can be expressed in terms of elementary or special functions. The main emphasis is on mastering the techniques, using the physical and geometric considerations, which often allow a considerable part of the results to be obtained without using complicated mathematical tools