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

TL;DR

This paper explores Lie-Hamilton systems, revealing their algebraic structure and providing methods to derive constants of motion and superposition rules, with applications to various differential equations.

Contribution

It introduces a Poisson coalgebra framework for Lie-Hamilton systems, enabling algebraic derivation of constants of motion and superposition rules.

Findings

Lie-Hamilton systems have a natural Poisson coalgebra structure.

Methods to derive constants of motion algebraically are developed.

Applications to Kummer-Schwarz, Riccati, Ermakov, and Smorodinsky-Winternitz systems.

Abstract

A Lie system is a nonautonomous 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 to derive 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.