A representation-theoretical approach to higher-dimensional Lie-Hamilton systems: The symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{R})$
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| Format: | Preprint |
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2024
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| _version_ | 1866910711820582912 |
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| author | Campoamor-Stursberg, Rutwig Carballal, Oscar Herranz, Francisco J. |
| author_facet | Campoamor-Stursberg, Rutwig Carballal, Oscar Herranz, Francisco J. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_17479 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A representation-theoretical approach to higher-dimensional Lie-Hamilton systems: The symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{R})$ Campoamor-Stursberg, Rutwig Carballal, Oscar Herranz, Francisco J. Mathematical Physics Dynamical Systems Exactly Solvable and Integrable Systems 34A26, 34C14, 17B10, 58A30 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. |
| title | A representation-theoretical approach to higher-dimensional Lie-Hamilton systems: The symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{R})$ |
| topic | Mathematical Physics Dynamical Systems Exactly Solvable and Integrable Systems 34A26, 34C14, 17B10, 58A30 |
| url | https://arxiv.org/abs/2406.17479 |