Seminarium KMMF "Teoria Dwoistości"

The main aim of this talk is to 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. First, we review the local classification of finite-dimensional real Lie algebras of vector fields on the plane. 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. As an application of our results, we investigate new and known Lie–Hamilton systems appearing in physical and mathematical problems: the Milne–Pinney, second-order Kummer–Schwarz, Cayley–Klein Riccati and Buchdahl equations as well as some Lotka–Volterra and other nonlinear biomathematical models.