A representation-theoretical approach to higher-dimensional Lie-Hamilton systems: The symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{R})$

TL;DR

This paper introduces a representation-theoretical method to construct higher-dimensional Lie-Hamilton systems, including new four-dimensional systems from $rak{sp}(4,b R)$, with applications to generalized oscillators and electromagnetic systems.

Contribution

It develops a novel Lie algebra representation approach for higher-dimensional Lie-Hamilton systems and introduces the concept of intrinsic systems, providing new models and superposition rules.

Findings

Constructed new four-dimensional Lie-Hamilton systems from $rak{sp}(4,b R)$

Identified intrinsic properties of these systems and related subalgebras

Derived superposition rules using the coalgebra method

Abstract

A new procedure for the construction of higher-dimensional Lie-Hamilton systems is proposed. This method is based on techniques belonging to the representation theory of Lie algebras and their realization by vector fields. The notion of intrinsic Lie-Hamilton system is defined, and a sufficiency criterion for this property given. Novel four-dimensional Lie-Hamilton systems arising from the fundamental representation of the symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{R})$ are obtained and proved to be intrinsic. Two distinguished subalgebras, the two-photon Lie algebra $\mathfrak{h}_{6}$ and the Lorentz Lie algebra $\mathfrak{so}(1,3)$, are also considered in detail. As applications, coupled time-dependent systems which generalize the Bateman oscillator and the one-dimensional Caldirola-Kanai models are constructed, as well as systems depending on a time-dependent electromagnetic field and generalized coupled oscillators. A superposition rule for these systems, exhibiting interesting symmetry properties, is obtained using the coalgebra method.