Lie–Hamilton systems on the plane: applications and superposition rules

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A Lie–Hamilton (LH) system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of Hamiltonian vector fields with respect to a Poisson structure. We provide new algebraic/geometric techniques to easily determine the properties of such Lie algebras on the plane, e.g., their associated Poisson bivectors. We study new and known LH systems on with physical, biological and mathematical applications. New results cover Cayley–Klein Riccati equations, the here defined planar diffusion Riccati systems, complex Bernoulli differential equations and projective Schrödinger equations. Constants of motion for planar LH systems are explicitly obtained which, in turn, allow us to derive superposition rules through a coalgebra approach.