Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Camassa-Holm equation as a geodesic flow for the $H^1$
  right-invariant metric   The fundamental role played by the Lie groups in mechanics, and especially by
the dual space of the Lie algebra of the group and the coadjoint action are
illustrated through the Camassa-Holm equation (CH). In 1996 Misio{\l}ek
observed that CH is a geodesic flow equation on the group of diffeomorphisms,
preserving the $H^1$ metric. This example is analogous to the Euler equations
in hydrodynamics, which describe geodesic flow for a right-invariant metric on
the infinite-dimensional group of diffeomorphisms preserving the volume element
of the domain of fluid flow and to the Euler equations of rigid body whith a
fixed point, describing geodesics for a left-invariant metric on SO(3). The
momentum map and an explicit parametrization of the Virasoro group, related to
recently obtained solutions for the CH equation are presented.

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The categories of this paper are: cmp-lg
18
The categories of this paper are: econ
10
The categories of this paper are: q-bio
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The categories of this paper are: plasm-ph
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The categories of this paper are: supr-con
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The categories of this paper are: solv-int
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The categories of this paper are: q-fin
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The categories of this paper are: eess
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The categories of this paper are: dg-ga
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The categories of this paper are: funct-an
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-th
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The categories of this paper are: adap-org
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The categories of this paper are: mtrl-th
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The categories of this paper are: chao-dyn