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arXiv · 2406.17479

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

Rutwig Campoamor-Stursberg·Oscar Carballal·Francisco J. Herranz
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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.

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Rutwig Campoamor-Stursberg, Oscar Carballal, Francisco J. Herranz. 2024-06-25. A representation-theoretical approach to higher-dimensional Lie-Hamilton systems: The symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{R})$. https://doi.org/10.1016/j.cnsns.2024.108452

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