Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 074, 12 pages      arXiv:2103.10495

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

Tomasz Stachowiak a and 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

Received May 04, 2021, in final form July 27, 2021; Published online July 31, 2021

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.

Key words: Kepler problem; two-body problem; Heisenberg group; differential Galois group; integrability; sub-Riemannian manifold.

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