Geometric Models for Lie–Hamilton Systems on ℝ²

Abstract

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.

Keywords

• lie system
• superposition rule
• lie–hamilton system
• integral system
• symplectic geometry
• lie algebra of vector fields