Stationary self-consistent distributions for a charged particle beam in the longitudinal magnetic field
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ationary Self-Consistent Distributions for a Charged Particle Beam in the Longitudinal Magnetic Field O. I. Drivotin and D. A. Ovsyannikov St. Petersburg State University, St. Petersburg, 199034 Russia e-mail: [email protected] Abstract―A review of analytical solutions of the Vlasov equation for a beam of charged particles is given. These results are analyzed on the basis of a unified approach developed by the authors. In the context of this method, a space of integrals of motion is introduced in which the integrals of motion of particles are considered as coordinates. In this case, specifying a self-consistent distribution is reduced to defining a distribution density in this space. This approach allows us to simplify the construction and analysis of different self-consistent distributions. In particular, it is possible, in some cases, to derive new solutions by considering linear combinations of well-known solutions. This approach also makes it possible in many cases to give a visual geometric representation of self-consistent distributions in the space of integrals of motion. DOI: 10.1134/S1063779616050038
1. INTRODUCTION One of the best known equations of mathematical physics is the Vlasov equation which is an integro-differential equation with partial derivatives [1–5]. The Vlasov equation is widely used in physics of charged particle beams for describing the evolution of particle distributions in the phase space [6–15]. Solutions of the Vlasov equation are called the self-consistent distributions, since in the context of the Vlasov equation, the effect of the field (which is called the self-field or self-consistent field), created by the ensemble of moving particles, on itself is taken into account. Issues of existence and uniqueness of solutions to the Vlasov equation were considered in [16–27]. Finding solutions to the Vlasov equation is a very difficult problem from the mathematical viewpoint, which is due to the nonlinear nature of the Vlasov equation. A large number of works have been devoted to this problem, in which different solutions to the Vlasov equation are found. The best known solution for a beam of charged particles both in the channel with quadrupole focusing by an electric field and in the channel with focusing by a longitudinal magnetic field is the Kapchinskii–Vladimirskii distribution [7, 15, 28]. The Kapchinskii–Vladimirskii distribution is an exact solution to the Vlasov equation. Solutions to the Vlasov equation are also sought in the form of a certain series (e.g., [29]). This work considers stationary self-consistent distributions for a charged particle beam in a longitudinal magnetic field. Different exact solutions are also wellknown here, e.g., the “water bag” distribution [15, 30].
The goal of this work is to review and analyze these results on the basis of a unified approach developed by the authors [31–45]. Within this approach, the integrals of motion of particles are considered as coordinates in the space of these integrals, while the specification of a self-consistent distri
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