Tomasz Stachowiak, Andrzej J. Maciejewski, “Non-Integrability of the Kepler and the Two-Body Problems on the Heisenberg Group”, SIGMA, 2021, Volume 17,074, 12 pp.

This article is cited in 1 paper

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

Tomasz Stachowiak^a, Andrzej J. Maciejewski^b

^a Kraków, Poland
^b Janusz Gil Institute of Astronomy, University of Zielona Góra, Licealna 9, PL-65–417 Zielona Góra, Poland

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.

MSC: 37J30, 70F05, 70H07, 70G45, 53C17

Received: May 4, 2021; in final form July 27, 2021; Published online July 31, 2021

Language: English

DOI: 10.3842/SIGMA.2021.074