Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Shape Dynamics of $N$ Point Vortices on the Sphere   We give a geometric account of the relative motion or the shape dynamics of
$N$ point vortices on the sphere exploiting the $\mathsf{SO}(3)$-symmetry of
the system. The main idea is to bypass the technical difficulty of the
$\mathsf{SO}(3)$-reduction by first lifting the dynamics from $\mathbb{S}^{2}$
to $\mathbb{C}^{2}$. We then perform the $\mathsf{U}(2)$-reduction using a dual
pair to obtain a Lie--Poisson dynamics for the shape dynamics. This
Lie--Poisson structure helps us find a family of Casimirs for the shape
dynamics. We further reduce the system by $\mathbb{T}^{N-1}$-symmetry to obtain
a Poisson structure for the shape dynamics involving less shape variables than
those of the previous work by Borisov and Pavlov. As an application of the
shape dynamics, we prove that the tetrahedron relative equilibria are stable
when all of their circulations have the same sign, generalizing some existing
results on tetrahedron relative equilibria of identical vortices.

20
The categories of this paper are: acc-phys
27
The categories of this paper are: comp-gas
1
The categories of this paper are: cs
33
The categories of this paper are: plasm-ph
35
The categories of this paper are: mtrl-th
36
The categories of this paper are: cmp-lg
6
The categories of this paper are: astro-ph
4
The categories of this paper are: cond-mat
28
The categories of this paper are: alg-geom
16
The categories of this paper are: stat
5
The categories of this paper are: gr-qc
8
The categories of this paper are: hep-ex
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The categories of this paper are: hep-lat
12
The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio
24
The categories of this paper are: dg-ga