Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Non-commutative integrability, exact solvability and the Hamilton-Jacobi
  theory   The non-commutative integrability (NCI) is a property fulfilled by some
Hamiltonian systems that ensures, among other things, the exact solvability of
their corresponding equations of motion. The latter means that an "explicit
formula" for the trajectories of these systems can be constructed. Such a
construction rests mainly on the so-called Lie theorem on integrability by
quadratures. It is worth mentioning that, in the context of Hamiltonian
systems, the NCI has been for around 40 years, essentially, the unique
criterium for exact solvability expressed in the terms of first integrals
(containing the usual Liouville-Arnold integrability criterium as a particular
case). Concretely, a Hamiltonian system with $n$ degrees of freedom is said to
be non-commutative integrable if a set of independent first integrals
$F_{1},...,F_{l}$ are known such that: the kernel of the $l\times l$ matrix
with coefficients $\left\{ F_{i},F_{j}\right\} $, where $\left\{
\cdot,\cdot\right\} $ denotes the canonical Poisson bracket, has dimension
$2n-l$ (isotropy); and each bracket $\left\{ F_{i},F_{j}\right\} $ is
functionally dependent on $F_{1},...,F_{l}$ (closure). In this paper, we
develop two procedures for constructing the trajectories of a Hamiltonian
system which only require isotropic first integrals (closure condition is not
needed). One of them is based on an extended version of the geometric
Hamilton-Jacobi theory, and does not rely on the above mentioned Lie's theorem.
We do all that in the language of functions of several variables.

18
The categories of this paper are: econ
23
The categories of this paper are: solv-int
20
The categories of this paper are: acc-phys
36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph
32
The categories of this paper are: atom-ph
21
The categories of this paper are: adap-org
11
The categories of this paper are: hep-lat
30
The categories of this paper are: q-alg
35
The categories of this paper are: mtrl-th
33
The categories of this paper are: plasm-ph
1
The categories of this paper are: cs
26
The categories of this paper are: bayes-an
13
The categories of this paper are: quant-ph