Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Analytic Central Orbits and their Transformation Group   A useful crude approximation for Abelian functions is developed and applied
to orbits. The bound orbits in the power-law potentials A*r^{-alpha} take the
simple form (l/r)^k = 1 + e cos(m*phi), where k = 2 - alpha > 0 and 'l' and 'e'
are generalisations of the semi-latus-rectum and the eccentricity. 'm' is given
as a function of 'eccentricity'. For nearly circular orbits 'm' is sqrt{k},
while the above orbit becomes exact at the energy of escape where 'e' is one
and 'm' is 'k'. Orbits in the logarithmic potential that gives rise to a
constant circular velocity are derived via the limit of small alpha. For such
orbits, r^2 vibrates almost harmonically whatever the 'eccentricity'. Unbound
orbits in power-law potentials are given in an appendix. The transformation of
orbits in one potential to give orbits in a different potential is used to
determine orbits in potentials that are positive powers of r. These
transformations are extended to form a group which associates orbits in sets of
six potentials, e.g. there are corresponding orbits in the potentials
proportional to r, r^{-2/3}, r^{-3}, r^{-6}, r^{4/3} and r^{-4}. A degeneracy
reduces this to three, which are r^{-1}, r^2 and r^{-4} for the Keplerian case.
A generalisation of this group includes the isochrone with the Kepler set.

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The categories of this paper are: q-bio
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The categories of this paper are: funct-an
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The categories of this paper are: alg-geom
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The categories of this paper are: quant-ph
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The categories of this paper are: math
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The categories of this paper are: supr-con
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The categories of this paper are: atom-ph
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The categories of this paper are: cs
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The categories of this paper are: comp-gas
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The categories of this paper are: patt-sol
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The categories of this paper are: dg-ga
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The categories of this paper are: solv-int
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The categories of this paper are: gr-qc
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The categories of this paper are: acc-phys
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The categories of this paper are: cond-mat
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The categories of this paper are: nucl-ex