Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Lie-Poisson structure of the LAE-$\alpha$ equation   This paper shows that the time $t$ map of the averaged Euler equations, with
Dirichlet, Neumann, and mixed boundary conditions is canonical relative to a
Lie-Poisson bracket constructed via a non-smooth reduction for the
corresponding diffeomorphism groups. It is also shown that the geodesic spray
for Neumann and mixed boundary conditions is smooth, a result already known for
Dirichlet boundary conditions.

10
The categories of this paper are: q-bio
22
The categories of this paper are: chao-dyn
6
The categories of this paper are: astro-ph
35
The categories of this paper are: mtrl-th
18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
1
The categories of this paper are: cs
21
The categories of this paper are: adap-org
9
The categories of this paper are: nlin
30
The categories of this paper are: q-alg
4
The categories of this paper are: cond-mat
0
The categories of this paper are: hep-ph
17
The categories of this paper are: q-fin
3
The categories of this paper are: physics
19
The categories of this paper are: eess
8
The categories of this paper are: hep-ex