Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Symplectic Reduction and the Lie--Poisson Shape Dynamics of $N$ Point
  Vortices on the Plane   We show that the symplectic reduction of the dynamics of $N$ point vortices
on the plane by the special Euclidean group $\mathsf{SE}(2)$ yields a
Lie--Poisson equation for relative configurations of the vortices.
Specifically, we combine symplectic reduction by stages with a dual pair
associated with the reduction by rotations to show that the
$\mathsf{SE}(2)$-reduced space with non-zero angular impulse is a coadjoint
orbit. This result complements some existing works by establishing a
relationship between the symplectic/Hamiltonian structures of the original and
reduced dynamics. We also find a family of Casimirs associated with the
Lie--Poisson structure including some apparently new ones. We demonstrate
through examples that one may exploit these Casimirs to show that some shape
dynamics are periodic.

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The categories of this paper are: bayes-an
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The categories of this paper are: cond-mat
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: stat
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The categories of this paper are: chem-ph
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The categories of this paper are: supr-con
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The categories of this paper are: econ
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The categories of this paper are: gr-qc
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The categories of this paper are: ao-sci
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-bio
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The categories of this paper are: adap-org
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The categories of this paper are: comp-gas