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arXiv · 1911.01094

Geometric models for Lie--Hamilton systems on $\mathbb{R}^2$

J. Lange·J. de Lucas
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Abstract

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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BibTeXRIS

J. Lange, J. de Lucas. 2019-11-04. Geometric models for Lie--Hamilton systems on $\mathbb{R}^2$. https://doi.org/10.3390/math7111053

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