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Geometric Models for Lie–Hamilton Systems on ℝ2

Department of Mathematical Methods in Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland
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Mathematics 2019, 7(11), 1053; https://doi.org/10.3390/math7111053
Received: 19 September 2019 / Revised: 24 October 2019 / Accepted: 29 October 2019 / Published: 4 November 2019
This paper provides a geometric description for Lie–Hamilton systems on 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 a natural framework for the analysis of Lie–Hamilton systems on 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. View Full-Text
Keywords: Lie system; superposition rule; Lie–Hamilton system; integral system; symplectic geometry; Lie algebra of vector fields Lie system; superposition rule; Lie–Hamilton system; integral system; symplectic geometry; Lie algebra of vector fields
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Lange, J.; de Lucas, J. Geometric Models for Lie–Hamilton Systems on ℝ2. Mathematics 2019, 7, 1053.

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