Geometric Models for Lie–Hamilton Systems on ℝ2

Julia Lange,Javier De Lucas

doi:10.3390/math7111053

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.

Highlights

A Lie system is a first-order system of ordinary differential equations (ODEs)whose general solution can be written as a function, a so-called superposition rule, of a generic family of particular solutions and some constants to be related to initial conditions [1,2,3,4,5,6,7]
We prove that the method given in Section 3 can be extended to the Lie algebras iso2 and iso1,1 to obtain Lie–Hamilton systems on their symplectic leaves related to VG Lie algebras locally diffeomorphic to elements of the class I14A for r = 2
We have developed two natural geometric models for the study of Lie–Hamilton systems on the plane that, quite surely, can be used to study higher-dimensional Lie–Hamilton systems

Summary

Introduction

A Lie system is a first-order system of ordinary differential equations (ODEs)whose general solution can be written as a function, a so-called superposition rule, of a generic family of particular solutions and some constants to be related to initial conditions [1,2,3,4,5,6,7]. Only twelve classes can be considered, again locally around a generic point, as VG Lie algebras of Hamiltonian vector fields relative to a symplectic form Many works on Lie–Hamilton systems try to derive or to explain the existence of a Poisson bivector or symplectic form turning the elements of a VG Lie algebra into Hamiltonian vector fields [10,19]. This has been done by solving systems of partial differential equations (PDEs) [12] or using other algebraic and geometric techniques [10,14,19]. VG Lie algebras of right-invariant vector fields and associated left-invariant tensor fields

Fundamentals

Lie–Hamilton Systems on R2 Related to Simple VG Lie Algebras

Non-Simple VG Lie Algebras and Symplectic Foliations

Conclusions and Outlook