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

SIGMA 17 (2021), 074, 12 pages      arXiv:2103.10495      https://doi.org/10.3842/SIGMA.2021.074

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

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.

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

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References

  1. Arnold V.I., Kozlov V.V., Neishtadt A.I., Mathematical aspects of classical and celestial mechanics, 3rd ed., Encyclopaedia of Mathematical Sciences, Vol. 3, Springer-Verlag, Berlin, 2006.
  2. Borisov A.V., Mamaev I.S., Bizyaev I.A., The spatial problem of 2 bodies on a sphere. Reduction and stochasticity, Regul. Chaotic Dyn. 21 (2016), 556-580.
  3. Compoint E., Weil J.A., Absolute reducibility of differential operators and Galois groups, J. Algebra 275 (2004), 77-105.
  4. Dods V., Shanbrom C., Self-similarity in the Kepler-Heisenberg problem, J. Nonlinear Sci. 31 (2021), 49, 15 pages, arXiv:1912.12375.
  5. Duval A., Loday-Richaud M., Kovačič's algorithm and its application to some families of special functions, Appl. Algebra Engrg. Comm. Comput. 3 (1992), 211-246.
  6. Folland G.B., A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc. 79 (1973), 373-376.
  7. Kovacic J.J., An algorithm for solving second order linear homogeneous differential equations, J. Symbolic Comput. 2 (1986), 3-43.
  8. Maciejewski A.J., Przybylska M., Integrability of Hamiltonian systems with algebraic potentials, Phys. Lett. A 380 (2016), 76-82.
  9. Montgomery R., Shanbrom C., Keplerian dynamics on the Heisenberg group and elsewhere, in Geometry, Mechanics, and Dynamics, Fields Inst. Commun., Vol. 73, Springer, New York, 2015, 319-342, arXiv:1212.2713.
  10. Morales-Ruiz J.J., Ramis J.P., Galoisian obstructions to integrability of Hamiltonian systems. I, Methods Appl. Anal. 8 (2001), 33-95.
  11. Rehm H.P., Galois groups and elementary solutions of some linear differential equations, J. Reine Angew. Math. 307-308 (1979), 1-7.
  12. Shanbrom C., Periodic orbits in the Kepler-Heisenberg problem, J. Geom. Mech. 6 (2014), 261-278, arXiv:1311.6061.
  13. Singer M.F., Ulmer F., Necessary conditions for Liouvillian solutions of (third order) linear differential equations, Appl. Algebra Engrg. Comm. Comput. 6 (1995), 1-22.
  14. Walter W., Ordinary differential equations, Graduate Texts in Mathematics, Vol. 182, Springer-Verlag, New York, 1998.

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