Skip to main content
SpringerLink
Log in

Unifying the Hyperbolic and Spherical \(2\)-Body Problem with Biquaternions

  • Published:
Regular and Chaotic Dynamics Aims and scope Submit manuscript

Abstract

The \(2\)-body problem on the sphere and hyperbolic space are both real forms of holomorphic Hamiltonian systems defined on the complex sphere. This admits a natural description in terms of biquaternions and allows us to address questions concerning the hyperbolic system by complexifying it and treating it as the complexification of a spherical system. In this way, results for the \(2\)-body problem on the sphere are readily translated to the hyperbolic case. For instance, we implement this idea to completely classify the relative equilibria for the \(2\)-body problem on hyperbolic 3-space and discuss their stability for a strictly attractive potential.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2.
Fig. 3

References

  1. Arathoon, P., Singular Reduction of the \(2\)-Body Problem on the \(3\)-Sphere and the \(4\)-Dimensional Spinning Top, Regul. Chaotic Dyn., 2019, vol. 24, no. 4, pp. 370–391.

    Article  MathSciNet  MATH  Google Scholar 

  2. Arathoon, P. and Fontaine, M., Real Forms of Holomorphic Hamiltonian Systems, arXiv:2009.10417 (2020).

  3. Borisov, A. V., García-Naranjo, L. C., Mamaev, I. S., and Montaldi, M., Reduction and Relative Equilibria for the Two-Body Problem on Spaces of Constant Curvature, Celestial Mech. Dynam. Astronom., 2018, vol. 130, no. 6, Art. 43, 36 pp.

    Article  MathSciNet  MATH  Google Scholar 

  4. Diacu, F., Pérez-Chavala, E., and Reyes Victoria, J. G., An Intrinsic Approach in the Curved \(n\)-Body Problem: The Negative Curvature Case, J. Differential Equations, 2012, vol. 252, no. 8, pp. 4529–4562.

    Article  MathSciNet  MATH  Google Scholar 

  5. Fulton, W. and Harris, J., Representation Theory: A First Course, Grad. Texts Math., vol. 129, New York: Springer, 2004.

    MATH  Google Scholar 

  6. García-Naranjo, L. C., Marrero, J. C., Pérez-Chavela, E., and Rodríguez-Olmos, M., Classification and Stability of Relative Equilibria for the Two-Body Problem in the Hyperbolic Space of Dimension \(2\), J. Differential Equations, 2016, vol. 260, no. 7, pp. 6375–6404.

    Article  MathSciNet  MATH  Google Scholar 

  7. Holm, D. D., Marsden, J. E., and Ratiu, T. S., The Euler – Poincaré Equations and Semidirect Products with Applications to Continuum Theories, Adv. Math., 1998, vol. 137, no. 1, pp. 1–81.

    Article  MathSciNet  MATH  Google Scholar 

  8. Marsden, J. E., Lectures on Mechanics, London Math. Soc. Lect. Note Ser., vol. 174, Cambridge: Cambridge Univ. Press, 1992.

    Book  MATH  Google Scholar 

  9. Marsden, J. E., Misiolek, G., Ortega, J.-P., Perlmutter, M., and Ratiu, T. S., Hamiltonian Reduction by Stages, Lecture Notes in Math., vol. 1913, Berlin: Springer, 2007.

    MATH  Google Scholar 

  10. O’Shea, L. and Sjamaar, R., Moment Maps and Riemannian Symmetric Pairs, Math. Ann., 2000, vol. 317, no. 3, pp. 415–457.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

At the time of writing the author was funded by an EPSRC Doctoral Prize Award hosted by the University of Manchester.

Author information

Authors and Affiliations

  1. University of Michigan, 2074 East Hall, 530 Church Street, MI 48109, Ann Arbor, USA

    Philip Arathoon

Authors
  1. Philip Arathoon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Philip Arathoon.

Ethics declarations

The author declares that he has no conflicts of interest.

Additional information

PUBLISHER’S NOTE

Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

MSC2010

70F05, 53D20

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arathoon, P. Unifying the Hyperbolic and Spherical \(2\)-Body Problem with Biquaternions. Regul. Chaot. Dyn. 28, 822–834 (2023). https://doi.org/10.1134/S1560354723060011

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1560354723060011

Keywords

Search

Navigation