Non-Integrability of the Kepler and the Two-Body Problems on the Heisenberg Group
The analog of the Kepler system defined on the Heisenberg group introduced by Montgomery and Shanbrom in [Fields Inst. Commun., Vol. 73, Springer, New York, 2015, 319-342, arXiv:1212.2713] is integrable on the zero level of the Hamiltonian. We show that in all other cases the system is not Liouville...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2021 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2021
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211349 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Non-Integrability of the Kepler and the Two-Body Problems on the Heisenberg Group. Tomasz Stachowiak and Andrzej J. Maciejewski. SIGMA 17 (2021), 074, 12 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862652428624592896 |
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| author | Stachowiak, Tomasz Maciejewski, Andrzej J. |
| author_facet | Stachowiak, Tomasz Maciejewski, Andrzej J. |
| citation_txt | Non-Integrability of the Kepler and the Two-Body Problems on the Heisenberg Group. Tomasz Stachowiak and Andrzej J. Maciejewski. SIGMA 17 (2021), 074, 12 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The analog of the Kepler system defined on the Heisenberg group introduced by Montgomery and Shanbrom in [Fields Inst. Commun., Vol. 73, Springer, New York, 2015, 319-342, arXiv:1212.2713] is integrable on the zero level of the Hamiltonian. We show that in all other cases the system is not Liouville integrable due to the lack of additional meromorphic first integrals. We prove that the analog of the two-body problem on the Heisenberg group is not integrable in the Liouville sense.
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| first_indexed | 2026-03-15T16:30:14Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211349 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T16:30:14Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Stachowiak, Tomasz Maciejewski, Andrzej J. 2025-12-30T15:53:36Z 2021 Non-Integrability of the Kepler and the Two-Body Problems on the Heisenberg Group. Tomasz Stachowiak and Andrzej J. Maciejewski. SIGMA 17 (2021), 074, 12 pages 1815-0659 2020 Mathematics Subject Classification: 37J30; 70F05; 70H07; 70G45; 53C17 arXiv:2103.10495 https://nasplib.isofts.kiev.ua/handle/123456789/211349 https://doi.org/10.3842/SIGMA.2021.074 The analog of the Kepler system defined on the Heisenberg group introduced by Montgomery and Shanbrom in [Fields Inst. Commun., Vol. 73, Springer, New York, 2015, 319-342, arXiv:1212.2713] is integrable on the zero level of the Hamiltonian. We show that in all other cases the system is not Liouville integrable due to the lack of additional meromorphic first integrals. We prove that the analog of the two-body problem on the Heisenberg group is not integrable in the Liouville sense. We would like to thank the anonymous referees for helping to improve the manuscript. This work has been supported by grants No. DEC-2011/02/A/ST1/00208 and DEC-2013/09/B/ST1/04130 of the National Science Centre of Poland. For the second author, this research was partially supported by the National Science Center of Poland Under Grant No. 2020/39/D/ST1/01632. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Non-Integrability of the Kepler and the Two-Body Problems on the Heisenberg Group Article published earlier |
| spellingShingle | Non-Integrability of the Kepler and the Two-Body Problems on the Heisenberg Group Stachowiak, Tomasz Maciejewski, Andrzej J. |
| title | Non-Integrability of the Kepler and the Two-Body Problems on the Heisenberg Group |
| title_full | Non-Integrability of the Kepler and the Two-Body Problems on the Heisenberg Group |
| title_fullStr | Non-Integrability of the Kepler and the Two-Body Problems on the Heisenberg Group |
| title_full_unstemmed | Non-Integrability of the Kepler and the Two-Body Problems on the Heisenberg Group |
| title_short | Non-Integrability of the Kepler and the Two-Body Problems on the Heisenberg Group |
| title_sort | non-integrability of the kepler and the two-body problems on the heisenberg group |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211349 |
| work_keys_str_mv | AT stachowiaktomasz nonintegrabilityofthekeplerandthetwobodyproblemsontheheisenberggroup AT maciejewskiandrzejj nonintegrabilityofthekeplerandthetwobodyproblemsontheheisenberggroup |