Abstract
The Kepler–Heisenberg problem is that of determining the motion of a planet around a sun in the Heisenberg group, thought of as a three-dimensional sub-Riemannian manifold. The sub-Riemannian Hamiltonian provides the kinetic energy, and the gravitational potential is given by the fundamental solution to the sub-Laplacian. The dynamics are at least partially integrable, possessing two first integrals as well as a dilational momentum which is conserved by orbits with zero energy. The system is known to admit closed orbits of any rational rotation number, which all lie within the fundamental zero-energy integrable subsystem. Here, we demonstrate that, under mild conditions, zero-energy orbits are self-similar. Consequently, we find that these zero-energy orbits stratify into three families: future collision, past collision, and quasi-periodicity without collision. If a collision occurs, it occurs in finite time.
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Acknowledgements
The authors are grateful to Richard Montgomery for many valuable conversations. The first author would like to thank MSRI for the privilege of taking part in the Fall 2018 Hamiltonian Systems semester, during which the bulk of this project was completed.
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Communicated by Melvin Leok.
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This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the first author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during Fall 2018.
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Dods, V., Shanbrom, C. Self-similarity in the Kepler–Heisenberg Problem. J Nonlinear Sci 31, 49 (2021). https://doi.org/10.1007/s00332-021-09709-1
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DOI: https://doi.org/10.1007/s00332-021-09709-1