Non-integrability of the Kepler and the two-body problem on the Heisenberg group

TL;DR

This paper proves that the Kepler and two-body problems on the Heisenberg group are generally not integrable in the Liouville sense, except at the zero Hamiltonian level, highlighting limitations of integrability in this geometric setting.

Contribution

It demonstrates the non-integrability of the Kepler and two-body problems on the Heisenberg group beyond the zero Hamiltonian case, extending understanding of dynamical systems on this geometry.

Findings

Kepler system on Heisenberg group is integrable only at zero Hamiltonian

Two-body problem on Heisenberg group is not Liouville integrable

No additional meromorphic first integrals exist in general

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.