Self-similarity in the Kepler–Heisenberg Problem

Dods, Victor; Shanbrom, Corey

doi:10.1007/s00332-021-09709-1

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2021

2024

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Cited by 2 publications

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“…where κ is a non-zero real parameter. The properties of this system have been analysed in [4,9,12], where it was shown that it has an invariant submanifold given by z = 0, p z = 0, p θ = xp y − yp x = 0 on which all trajectories are straight lines. More importantly, it was also demonstrated that all periodic solutions must lie on the zero-energy level and that the whole system is Liouville integrable there.…”

Section: Introductionmentioning

confidence: 99%

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

Maciejewski¹,

Stachowiak²

2021

SIGMA

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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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Section: Introductionmentioning

confidence: 99%

“…Remark 1.1. The form of potential in (1.2) is the same as in [4] which is the correction of that in [9].…”

Section: Introductionmentioning

confidence: 99%