Lie-Hamilton systems on the plane: properties, classification and applications

摘要

We study Lie-Hamilton systems on the plane, i.e. systems of first-orderdifferential equations describing the integral curves of a $t$-dependent vectorfield taking values in a finite-dimensional real Lie algebra of planarHamiltonian vector fields with respect to a Poisson structure. We start withthe local classification of finite-dimensional real Lie algebras of vectorfields on the plane obtained in [A. Gonz\'alez-L\'opez, N. Kamran and P.J.Olver, Proc. London Math. Soc. 64, 339 (1992)] and we interpret their resultsas a local classification of Lie systems. Moreover, by determining which ofthese real Lie algebras consist of Hamiltonian vector fields with respect to aPoisson structure, we provide the complete local classification of Lie-Hamiltonsystems on the plane. We present and study through our results new Lie-Hamiltonsystems of interest which are used to investigate relevant non-autonomousdifferential equations, e.g. we get explicit local diffeomorphisms between suchsystems. In particular, the Milne-Pinney, second-order Kummer-Schwarz, complexRiccati and Buchdahl equations as well as some Lotka-Volterra and nonlinearbiomathematical models are analysed from this Lie-Hamilton approach.