Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The evolution of the phase space density of particle beams in external
  fields   The evolution of the phase space density of particle beams in external fields
is found proceeding from the continuity equation in the six-dimensional (6D)
phase space (mu-space). The Robinson theorem, which includes the Liouville
theorem as a special case, was proved in a more simple and consistent
alternative way valid for arbitrary external fields, averaged fields of the
beam (self-generated electro-magnetic fields except intrabeam scattering) and
arbitrary frictional forces (linear, nonlinear). It includes particle
accelerators as a special case. The limits of the applicability of the Robinson
theorem in case of cooling of excited ions having a finite living time are
presented.

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The categories of this paper are: cmp-lg
23
The categories of this paper are: solv-int
9
The categories of this paper are: nlin
29
The categories of this paper are: funct-an
14
The categories of this paper are: math-ph
12
The categories of this paper are: nucl-th
2
The categories of this paper are: math
21
The categories of this paper are: adap-org
17
The categories of this paper are: q-fin
13
The categories of this paper are: quant-ph
26
The categories of this paper are: bayes-an
20
The categories of this paper are: acc-phys
22
The categories of this paper are: chao-dyn
25
The categories of this paper are: patt-sol
16
The categories of this paper are: stat
24
The categories of this paper are: dg-ga