New classes of stationary solutions of vlasov's equation for an axially symmetrical beam of charged particles of constant density

Modeling of self-consistent distributions for longitudinally non-uniform beams

“…Distributions were also found [4,6] for which the phase density depends on a linear combination of the energy of the transverse motion and the angular momentum of a particle. Wide classes of selfconsistent distributions have also been reported in the literature [7,8].…”

Covariant description of phase space distributions

“…Санкт-Петербургский государственный университет, Российская Федерация, 199034, Санкт-Петербург, Университетская наб., [7][8][9] Сформулирована концепция фазового пространства для частиц, движущихся в 4-мерном пространстве-времени. Дано определение плотности распределения частиц как дифференциальной формы.…”

“…In fact, such approach was used in numerous works devoted to self-consistent distributions for a cylindrical beam and for a beam with varying cross-section radius ("breathing" beam) [8][9][10][11][12][13][14][15]. In these works, the space of motion integrals was introduced, and particle density was defined in this space.…”

“…It gave possibility to construct new solutions of the Vlasov equation. For example, they can be obtained as a linear combinations of degenerate distributions, or as solutions of some integral equation [9][10][11][12][13][14][15].…”

“…One of the purposes of this article is to write down such equation. Therefore, formalization of the approach applied in works [8][9][10][11][12][13][14][15] is offered in this article.…”