Contact Lie systems on Riemannian and Lorentzian spaces: from scaling symmetries to curvature-dependent reductions
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| Format: | Preprint |
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2025
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| author | Campoamor-Stursberg, Rutwig Carballal, Oscar Herranz, Francisco J. |
| author_facet | Campoamor-Stursberg, Rutwig Carballal, Oscar Herranz, Francisco J. |
| contents | We propose an adaptation of the notion of scaling symmetries for the case of Lie-Hamilton systems, allowing their subsequent reduction to contact Lie systems. As an illustration of the procedure, time-dependent frequency oscillators and time-dependent thermodynamic systems are analyzed from this point of view. The formalism provides a novel method for constructing contact Lie systems on the three-dimensional sphere, derived from recently established Lie-Hamilton systems arising from the fundamental four-dimensional representation of the symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{R})$. It is shown that these systems are a particular case of a larger hierarchy of contact Lie systems on a special class of three-dimensional homogeneous spaces, namely the Cayley-Klein spaces. These include Riemannian spaces (sphere, hyperbolic and Euclidean spaces), pseudo-Riemannian spaces (anti-de Sitter, de Sitter and Minkowski spacetimes), as well as Newtonian or non-relativistic spacetimes. Under certain topological conditions, some of these systems retrieve well-known two-dimensional Lie-Hamilton systems through a curvature-dependent reduction. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_20558 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Contact Lie systems on Riemannian and Lorentzian spaces: from scaling symmetries to curvature-dependent reductions Campoamor-Stursberg, Rutwig Carballal, Oscar Herranz, Francisco J. Mathematical Physics Differential Geometry Symplectic Geometry 37J55, 34A26 (Primary), 17B66, 34C14, 70G45 (Secondary) We propose an adaptation of the notion of scaling symmetries for the case of Lie-Hamilton systems, allowing their subsequent reduction to contact Lie systems. As an illustration of the procedure, time-dependent frequency oscillators and time-dependent thermodynamic systems are analyzed from this point of view. The formalism provides a novel method for constructing contact Lie systems on the three-dimensional sphere, derived from recently established Lie-Hamilton systems arising from the fundamental four-dimensional representation of the symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{R})$. It is shown that these systems are a particular case of a larger hierarchy of contact Lie systems on a special class of three-dimensional homogeneous spaces, namely the Cayley-Klein spaces. These include Riemannian spaces (sphere, hyperbolic and Euclidean spaces), pseudo-Riemannian spaces (anti-de Sitter, de Sitter and Minkowski spacetimes), as well as Newtonian or non-relativistic spacetimes. Under certain topological conditions, some of these systems retrieve well-known two-dimensional Lie-Hamilton systems through a curvature-dependent reduction. |
| title | Contact Lie systems on Riemannian and Lorentzian spaces: from scaling symmetries to curvature-dependent reductions |
| topic | Mathematical Physics Differential Geometry Symplectic Geometry 37J55, 34A26 (Primary), 17B66, 34C14, 70G45 (Secondary) |
| url | https://arxiv.org/abs/2503.20558 |