Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Geometric models for Lie--Hamilton systems on $\mathbb{R}^2$   This paper provides a geometric description for Lie--Hamilton systems on
$\mathbb{R}^2$ with locally transitive Vessiot--Guldberg Lie algebras through
two types of geometric models. The first one is the restriction of a class of
Lie--Hamilton systems on the dual of a Lie algebra to even-dimensional
symplectic leaves relative to the Kirillov-Kostant-Souriau bracket. The second
is a projection onto a quotient space of an automorphic Lie--Hamilton system
relative to a naturally defined Poisson structure or, more generally, an
automorphic Lie system with a compatible bivector field. These models give rise
to a natural framework for the analysis of Lie--Hamilton systems on
$\mathbb{R}^2$ while retrieving known results in a natural manner. Our methods
may be extended to study Lie--Hamilton systems on higher-dimensional manifolds
and provide new approaches to Lie systems admitting compatible geometric
structures.

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The categories of this paper are: quant-ph
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The categories of this paper are: adap-org
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The categories of this paper are: dg-ga
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The categories of this paper are: alg-geom
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The categories of this paper are: plasm-ph
20
The categories of this paper are: acc-phys
3
The categories of this paper are: physics
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The categories of this paper are: q-bio
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The categories of this paper are: funct-an
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The categories of this paper are: q-fin
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The categories of this paper are: mtrl-th
34
The categories of this paper are: chem-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: cmp-lg
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The categories of this paper are: astro-ph
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The categories of this paper are: q-alg