Integrability of homogeneous exact magnetic flows on spheres
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912501881372672 |
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| author | Dragovic, Vladimir Gajic, Borislav Jovanovic, Bozidar |
| author_facet | Dragovic, Vladimir Gajic, Borislav Jovanovic, Bozidar |
| contents | We consider motion of a material point placed in a constant homogeneous magnetic field in $\mathbb R^n$ and also motion restricted to the sphere $S^{n-1}$. While there is an obvious integrability of the magnetic system in $\mathbb R^n$, the integrability of the system restricted to the sphere $S^{n-1}$ is highly non-trivial. We prove complete integrability of the obtained restricted magnetic systems for $n\le 6$. The first integrals of motion of the magnetic flows on the spheres $S^{n-1}$, for $n=5$ and $n=6$, are polynomials of the degree $1$, $2$, and $3$ in momenta. We prove noncommutative integrability of the obtained magnetic flows for any $n\ge 7$ when the systems allow a reduction to the cases with $n\le 6$. We conjecture that the restricted magnetic systems on $S^{n-1}$ are integrable for all $n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_20515 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integrability of homogeneous exact magnetic flows on spheres Dragovic, Vladimir Gajic, Borislav Jovanovic, Bozidar Differential Geometry Exactly Solvable and Integrable Systems 37J35, 53D25 We consider motion of a material point placed in a constant homogeneous magnetic field in $\mathbb R^n$ and also motion restricted to the sphere $S^{n-1}$. While there is an obvious integrability of the magnetic system in $\mathbb R^n$, the integrability of the system restricted to the sphere $S^{n-1}$ is highly non-trivial. We prove complete integrability of the obtained restricted magnetic systems for $n\le 6$. The first integrals of motion of the magnetic flows on the spheres $S^{n-1}$, for $n=5$ and $n=6$, are polynomials of the degree $1$, $2$, and $3$ in momenta. We prove noncommutative integrability of the obtained magnetic flows for any $n\ge 7$ when the systems allow a reduction to the cases with $n\le 6$. We conjecture that the restricted magnetic systems on $S^{n-1}$ are integrable for all $n$. |
| title | Integrability of homogeneous exact magnetic flows on spheres |
| topic | Differential Geometry Exactly Solvable and Integrable Systems 37J35, 53D25 |
| url | https://arxiv.org/abs/2504.20515 |