Lie–Hamilton systems on the plane: Properties, classification and applications

Abstract

We study Lie–Hamilton systems on the plane, i.e. systems of first-order differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of planar Hamiltonian vector fields with respect to a Poisson structure. We start with the local classification of finite-dimensional real Lie algebras of vector fields on the plane obtained in González-López, Kamran, and Olver (1992) [23] and we interpret their results as a local classification of Lie systems. By determining which of these real Lie algebras consist of Hamiltonian vector fields relative to a Poisson structure, we provide the complete local classification of Lie–Hamilton systems on the plane. We present and study through our results new Lie–Hamilton systems of interest which are used to investigate relevant non-autonomous differential equations, e.g. we get explicit local diffeomorphisms between such systems. We also analyse biomathematical models, the Milne–Pinney equations, second-order Kummer–Schwarz equations, complex Riccati equations and Buchdahl equations.