| Paper | Title | Page |
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| TUPPB028 | Degenerate Solutions of the Vlasov Equation | 376 |
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The report deals with degenerate solutions of the Vlasov equation. By degenerate solution we mean a distribution which have support of dimension smaller than dimension of the phase space. Well known example is the Kapchinsky-Vladimirsky (KV) distribution, when particles are distributed on the 3-dimensional surface in the 4-dimensional phase space. We use covariant formulation of the Vlasov equation developed previously*. In traditional approach, the Vlasov equation is considered as integro-differential equation with partial derivatives on phase coordinates. Covariant approach means tensor formulation. For the covariant formulation of the Vlasov equation, we use such tensor object as the Lie derivative. According to the covariant approach, a degenerate solution is described by differential form which degree is equal to the dimension of its support. Main attention is paid to the KV distribution, which is described by the differential form of the third degree. It is demonstrated that the KV distribution satisfies to the Vlasov equation in covariant formulation. It is shown, how one can set initial partical positions in the phase space to simulate that distribution. Some other distributions are also considered. This work has theoretical as well as practical significance. Presented results can be applied for description and simulation of high-intensity beam.
O.I. Drivotin. Covariant Formulation of the Vlasov Equation. Proc. of IPAC 2011, San-Sebastian, Spain. http://accelconf.web.cern.ch/AccelConf/IPAC2011/papers/wepc114.pdf |
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