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 [A. González-López, N. Kamran and P.J. Olver, Proc. London Math. Soc. 64, 339 (1992)] and we interpret their results as a local classification of Lie systems. Moreover, by determining which of these real Lie algebras consist of Hamiltonian vector fields with respect 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. In particular, the Milne-Pinney, second-order Kummer-Schwarz, complex Riccati and Buchdahl equations as well as some Lotka-Volterra and nonlinear biomathematical models are analysed from this Lie-Hamilton approach.

Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations

“…v ∂ ∂v and the associated SODE Lie system is easily seen to be an equation of the Liouville type (in various applications, this type of SODE Lie system is also known as the generalized Buchdahl equation [4]):…”

“…Among the low dimensional Lie algebras, the case r 2 sl (2, R) plays a special role, as it is related to various of the most relevant and best studied cases of SODE Lie systems (see, e.g., [2,4,5] and the references therein).…”

“…For a long time considered as a particular technique for differential equations, the importance of Lie systems in physical applications, particularly in the context of integrable systems, as well as in control theory, has motivated extensive studies on the subject in the last few decades (see, e.g., [2,3]) that have led to natural generalizations of the notion of Lie systems [4][5][6]. These generalizations of the classical analytical formulation allow an elegant and effective description of Lie systems in terms of distributions in the sense of Fröbenius and the theory of fiber bundles, hence offering a much wider spectrum of applications, as well as their adaptation to quantum systems (the excellent treatise on the geometrical foundations of Lie systems [2] contains an extensive and updated list of references enumerating these applications).…”

k-Symplectic Lie systems: theory and applications

“…Systems of this type have already been studied by one of the authors of this work in [7]. We hereafter call these systems Lie-Lotka-Volterra systems.…”

“…Nevertheless, Lie systems appear in important physical and mathematical problems and enjoy relevant geometric properties [4,20,23,25,26,30,51,53], which strongly prompt their analysis. Some attention has lately been paid to Lie systems admitting a Vessiot-Guldberg Lie algebra of Hamiltonian vector fields with respect to several geometric structures [7,8,21]. Surprisingly, studying these particular types of Lie systems led to investigate much more Lie systems and applications than before.…”