Geodesic
Solutions of the Vlasov equation for charged particle beam in magnetic field
The Bulletin of Irkutsk State University. Series Mathematics, Tome 6 (2013) no. 4, pp. 2-22

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New approach based on covariant formulation of the Vlasov equation is presented. This approach allows to use various coordinates in the phase space, and to consider degenerate distributions. It is shown how to apply this approach to the problem of finding of solutions of the Vlasov equation for charged particle beam. Solutions obtained within the framework of this approach are presented.
Keywords: the Vlasov equation; self-consistent distributions; phase density; Kapchinsky–Vladimirsky distribution; Ermakov integral. 
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     author = {O. I. Drivotin and D. A. Ovsyannikov},
     title = {Solutions of the {Vlasov} equation for charged particle beam in magnetic field},
     journal = {The Bulletin of Irkutsk State University. Series Mathematics},
     pages = {2--22},
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O. I. Drivotin; D. A. Ovsyannikov. Solutions of the Vlasov equation for charged particle beam in magnetic field. The Bulletin of Irkutsk State University. Series Mathematics, Tome 6 (2013) no. 4, pp. 2-22. http://geodesic.mathdoc.fr/item/IIGUM_2013_6_4_a0/