Lie-Hamilton systems on the plane: Applications and superposition rules

TL;DR

This paper develops algebraic and geometric methods to analyze Lie-Hamilton systems on the plane, providing new results and superposition rules for various differential equations with applications across physics, biology, and mathematics.

Contribution

It introduces new techniques for studying Lie-Hamilton systems on the plane and derives superposition rules using a coalgebra approach, covering new classes of equations.

Findings

Explicit constants of motion for planar Lie-Hamilton systems

Superposition rules derived via coalgebra methods

Application to new and known differential equations like Riccati and Schrödinger types

Abstract

A Lie-Hamilton 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 Lie-Hamilton systems on $\mathbb{R}^2$ 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 Schr\"odinger equations. Constants of motion for planar Lie-Hamilton systems are explicitly obtained which, in turn, allow us to derive superposition rules through a coalgebra approach.