Two-dimensional classical superintegrable systems: polynomial algebra of integrals
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913906495062016 |
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| author | Escobar-Ruiz, A. M. Azuaje, R. Gordiano, J. C. |
| author_facet | Escobar-Ruiz, A. M. Azuaje, R. Gordiano, J. C. |
| contents | In this work, we investigate generic classical two-dimensional (2D) superintegrable Hamiltonian systems H, characterized by the existence of three functionally independent integrals of motion (I_0=H,I_1,I_2). Our main result, formulated and proved as a theorem, establishes that the set (I_0,I_1,I_2,I_12={I_1,I_2}) generates a four-dimensional polynomial algebra under the Poisson bracket. Unlike previous studies, this study describes a construction that neither depends on the additive separability of the Hamilton-Jacobi equation nor presupposes polynomial integrals of motion in the canonical momenta. Specifically, we prove an instrumental observation presented in [D. Bonatsos et al., PRA 50, 3700 (1994)] concerning deformed oscillator algebras in superintegrable systems. We apply the method to a variety of physically relevant examples, including the Kepler system, Holt potential, Smorodinsky-Winternitz potential, Fokas-Lagerstrom potential, the Higgs oscillator, and the non-separable Post-Winternitz system. In several cases, we explicitly derive the form of the classical trajectories y=y(x;I_0,I_1,I_2) using purely algebraic means. Moreover, by examining the conditions under which I_1=I_2=0, we identify and characterize special classes of trajectories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_17519 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Two-dimensional classical superintegrable systems: polynomial algebra of integrals Escobar-Ruiz, A. M. Azuaje, R. Gordiano, J. C. Mathematical Physics In this work, we investigate generic classical two-dimensional (2D) superintegrable Hamiltonian systems H, characterized by the existence of three functionally independent integrals of motion (I_0=H,I_1,I_2). Our main result, formulated and proved as a theorem, establishes that the set (I_0,I_1,I_2,I_12={I_1,I_2}) generates a four-dimensional polynomial algebra under the Poisson bracket. Unlike previous studies, this study describes a construction that neither depends on the additive separability of the Hamilton-Jacobi equation nor presupposes polynomial integrals of motion in the canonical momenta. Specifically, we prove an instrumental observation presented in [D. Bonatsos et al., PRA 50, 3700 (1994)] concerning deformed oscillator algebras in superintegrable systems. We apply the method to a variety of physically relevant examples, including the Kepler system, Holt potential, Smorodinsky-Winternitz potential, Fokas-Lagerstrom potential, the Higgs oscillator, and the non-separable Post-Winternitz system. In several cases, we explicitly derive the form of the classical trajectories y=y(x;I_0,I_1,I_2) using purely algebraic means. Moreover, by examining the conditions under which I_1=I_2=0, we identify and characterize special classes of trajectories. |
| title | Two-dimensional classical superintegrable systems: polynomial algebra of integrals |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2506.17519 |