From constants of motion to superposition rules for Lie–Hamilton systems

Abstract

A Lie system is a non-autonomous system of first-order differential equations possessing a superposition rule, i.e. a map expressing its general solution in terms of a generic finite family of particular solutions and some constants. Lie–Hamilton systems form a subclass of Lie systems whose dynamics is governed by a curve in a finite-dimensional real Lie algebra of functions on a Poisson manifold. It is shown that Lie–Hamilton systems are naturally endowed with a Poisson coalgebra structure. This allows us to devise methods for deriving in an algebraic way their constants of motion and superposition rules. We illustrate our methods by studying Kummer–Schwarz equations, Riccati equations, Ermakov systems and Smorodinsky–Winternitz systems with time-dependent frequency.