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Symmetry, Integrability and Geometry: Methods and Applications, 2021, Volume 17, 074, 12 pp.
DOI: https://doi.org/10.3842/SIGMA.2021.074 (Mi sigma1756) 		 
This article is cited in 1 scientific paper (total in 1 paper)

Non-Integrability of the Kepler and the Two-Body Problems on the Heisenberg Group

Tomasz Stachowiaka, Andrzej J. Maciejewskib

a Kraków, Poland
b Janusz Gil Institute of Astronomy, University of Zielona Góra, Licealna 9, PL-65–417 Zielona Góra, Poland
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DOI: https://doi.org/10.3842/SIGMA.2021.074
Abstract: 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.
Keywords: Kepler problem, two-body problem, Heisenberg group, differential Galois group, integrability, sub-Riemannian manifold.
Funding agency Grant number
National Science Centre (Narodowe Centrum Nauki) DEC-2011/02/A/ST1/00208
DEC-2013/09/B/ST1/04130
2020/39/D/ST1/01632
This work has been supported by grants No. DEC-2011/02/A/ST1/00208 and DEC-2013/09/B/ST1/04130 of 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.
Received: May 4, 2021; in final form July 27, 2021; Published online July 31, 2021
Bibliographic databases:
Document Type: Article
MSC: 37J30, 70F05, 70H07, 70G45, 53C17
Language: English
Citation: Tomasz Stachowiak, Andrzej J. Maciejewski, “Non-Integrability of the Kepler and the Two-Body Problems on the Heisenberg Group”, SIGMA, 17 (2021), 074, 12 pp.
Citation in format AMSBIB
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