The preceding review of basic plasma physics, hydrodynamics, and electrodynamics (with special attention to the Lorentz force and the phases between the E and the H fields of electromagnetic waves and how this was derived) was necessary to understand the nonlinear forces and ponderomotion. In order to resolve the numerous open questions regarding the assumptions about the ponderomotive potential described in Chapter 1, and unless we are to work with this while ignoring the complicated background, we have no choice but to go through these details. For easier reading of this text, we begin first with a derivation of the nonlinear force for a simplified case of perpendicular incidence and determine the result of the force. After this, we discuss the problems that led to a very general derivation of the hydrodynamic basis of the nonlinear forces, which in turn arrived at a clarification of the hydrodynamic two-fluid theory. This leads to a general derivation of the Maxwell stress tensor in a purely hydrodynamic way, whereas all of the other derivations used elastomechanics.
The elastomechanical method provided very limited results for slow temporal changes for fluids without dispersion and without dissipation (absorption), whereas the hydrodynamic method immediately covers the high-frequency case of plasma dispersion and dissipation. Nevertheless, this is not the whole story. The problems of single-particle motions, the resulting electric double layers, and more general numerical evaluations on the basis of a genuine two-fluid model will follow in the next chapter, with the addition of applications for laser accelerators, laser fusion, ion sources, and related industrial uses.
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