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

k-Symplectic Lie systems: theory and applications

“…This system is important due to the fact that their t -independent constants of motion are employed to obtain a superposition rule for Riccati equations [8]. Let us show first that this system is a Lie system…”

“…Nevertheless, Lie systems appear in important physical and mathematical problems and enjoy relevant geometric properties [4,20,23,25,26,30,51,53], which strongly prompt their analysis. Some attention has lately been paid to Lie systems admitting a Vessiot-Guldberg Lie algebra of Hamiltonian vector fields with respect to several geometric structures [7,8,21]. Surprisingly, studying these particular types of Lie systems led to investigate much more Lie systems and applications than before.…”

“…can be retrieved as a Casimir element of a real Lie algebra of Hamiltonian functions [8]. Moreover, this work introduced the study of Poisson co-algebra techniques to obtain superposition rules for these systems [8].…”

“…Moreover, this work introduced the study of Poisson co-algebra techniques to obtain superposition rules for these systems [8].…”

“…they form a k-symplectic structure [29]. Such systems are relevant as they describe Schwarzian equations [14] and coupled Riccati equations [8], which have applications in the theory of Lie systems, classical mechanics and other fields [16].…”