Nonlinear Lie-Hamilton systems: $t$-Dependent curved oscillators and Kepler-Coulomb Hamiltonians
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| Format: | Preprint |
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2025
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| author | Campoamor-Stursberg, Rutwig Herranz, Francisco J. de Lucas, Javier |
| author_facet | Campoamor-Stursberg, Rutwig Herranz, Francisco J. de Lucas, Javier |
| contents | The Lie-Hamilton approach for $t$-dependent Hamiltonians is extended to cover the so-called nonlinear Lie-Hamilton systems, which are no longer related to a linear $t$-dependent combination of a basis of a finite-dimensional Lie algebra of functions $\mathcal{W}$, but an arbitrary $t$-dependent function on $\mathcal{W}$. This novel formalism is accomplished through a detailed analysis of related structures, such as momentum maps and generalized distributions, together with the extension of the Poisson coalgebra method to a $t$-dependent frame, in order to systematize the construction of constants of the motion for nonlinear systems. Several relevant relations between nonlinear Lie-Hamilton systems, Lie-Hamilton systems, and collective Hamiltonians are analyzed. The new notions and tools are illustrated with the study of the harmonic oscillator, Hénon-Heiles systems and Painlevé trascendents within a $t$-dependent framework. In addition, the formalism is carefully applied to construct oscillators with a $t$-dependent frequency and Kepler-Coulomb systems with a $t$-dependent coupling constant on the $n$-dimensional sphere, Euclidean and hyperbolic spaces, as well as on some spaces of non-constant curvature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_13853 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonlinear Lie-Hamilton systems: $t$-Dependent curved oscillators and Kepler-Coulomb Hamiltonians Campoamor-Stursberg, Rutwig Herranz, Francisco J. de Lucas, Javier Mathematical Physics Dynamical Systems Exactly Solvable and Integrable Systems 34A26, 34A34 (Primary), 34C14, 17B66, 70G65 (Secondary) The Lie-Hamilton approach for $t$-dependent Hamiltonians is extended to cover the so-called nonlinear Lie-Hamilton systems, which are no longer related to a linear $t$-dependent combination of a basis of a finite-dimensional Lie algebra of functions $\mathcal{W}$, but an arbitrary $t$-dependent function on $\mathcal{W}$. This novel formalism is accomplished through a detailed analysis of related structures, such as momentum maps and generalized distributions, together with the extension of the Poisson coalgebra method to a $t$-dependent frame, in order to systematize the construction of constants of the motion for nonlinear systems. Several relevant relations between nonlinear Lie-Hamilton systems, Lie-Hamilton systems, and collective Hamiltonians are analyzed. The new notions and tools are illustrated with the study of the harmonic oscillator, Hénon-Heiles systems and Painlevé trascendents within a $t$-dependent framework. In addition, the formalism is carefully applied to construct oscillators with a $t$-dependent frequency and Kepler-Coulomb systems with a $t$-dependent coupling constant on the $n$-dimensional sphere, Euclidean and hyperbolic spaces, as well as on some spaces of non-constant curvature. |
| title | Nonlinear Lie-Hamilton systems: $t$-Dependent curved oscillators and Kepler-Coulomb Hamiltonians |
| topic | Mathematical Physics Dynamical Systems Exactly Solvable and Integrable Systems 34A26, 34A34 (Primary), 34C14, 17B66, 70G65 (Secondary) |
| url | https://arxiv.org/abs/2505.13853 |