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

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.

arXiv comment15 pages, 1 figure

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Cited by in corpus (1)

• Geometric numerical methods for Lie systems and their application in optimal control