Manifolds with a commutative and associative product structure that encodes superintegrable Hamiltonian systems
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909567903858688 |
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| author | Vollmer, Andreas |
| author_facet | Vollmer, Andreas |
| contents | We show that two natural and a priori unrelated structures encapsulate the same data, namely certain commutative and associative product structures and a class of superintegrable Hamiltonian systems. More precisely, consider a Euclidean space of dimension at least three, equipped with a commutative and associative product structure that satisfies the conditions of a Manin-Frobenius manifold, plus one additional compatibility condition. We prove that such a product structure encapsulates precisely the conditions of a so-called abundant structure. Such a structure provides the data needed to construct a family of second-order (maximally) superintegrable Hamiltonian systems of second order. We prove that all abundant superintegrable Hamiltonian systems on Euclidean space of dimension at least three arise in this way. As an example, we present the Smorodinski-Winternitz Hamiltonian system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_06418 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Manifolds with a commutative and associative product structure that encodes superintegrable Hamiltonian systems Vollmer, Andreas Differential Geometry Mathematical Physics 37J39, 53D45, 70G45, 37J35, 70H33 We show that two natural and a priori unrelated structures encapsulate the same data, namely certain commutative and associative product structures and a class of superintegrable Hamiltonian systems. More precisely, consider a Euclidean space of dimension at least three, equipped with a commutative and associative product structure that satisfies the conditions of a Manin-Frobenius manifold, plus one additional compatibility condition. We prove that such a product structure encapsulates precisely the conditions of a so-called abundant structure. Such a structure provides the data needed to construct a family of second-order (maximally) superintegrable Hamiltonian systems of second order. We prove that all abundant superintegrable Hamiltonian systems on Euclidean space of dimension at least three arise in this way. As an example, we present the Smorodinski-Winternitz Hamiltonian system. |
| title | Manifolds with a commutative and associative product structure that encodes superintegrable Hamiltonian systems |
| topic | Differential Geometry Mathematical Physics 37J39, 53D45, 70G45, 37J35, 70H33 |
| url | https://arxiv.org/abs/2411.06418 |