Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Hydrodynamic Exercise in Free Probability: Setting up Free Euler
  Equations   For the free probability analogue of Euclidean space endowed with the
Gaussian measure we apply the approach of Arnold to derive Euler equations for
a Lie algebra of non-commutative vector fields which preserve a certain trace.
We extend the equations to vector fields satisfying non-commutative smoothness
requirements. We introduce a cyclic vorticity and show that it satisfies a
vorticity equation and that it produces a family of conserved quantities.

14
The categories of this paper are: math-ph
22
The categories of this paper are: chao-dyn
9
The categories of this paper are: nlin
12
The categories of this paper are: nucl-th
18
The categories of this paper are: econ
25
The categories of this paper are: patt-sol
32
The categories of this paper are: atom-ph
37
The categories of this paper are: supr-con
33
The categories of this paper are: plasm-ph
8
The categories of this paper are: hep-ex
36
The categories of this paper are: cmp-lg
35
The categories of this paper are: mtrl-th
34
The categories of this paper are: chem-ph
15
The categories of this paper are: nucl-ex
7
The categories of this paper are: hep-th
31
The categories of this paper are: ao-sci