Skip to main content
Log in

Circular Non-collision Orbits for a Large Class of n-Body Problems

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

We prove for a large class of n-body problems including a subclass of quasihomogeneous n-body problems, the classical n-body problem, the n-body problem in spaces of negative constant Gaussian curvature and a restricted case of the n-body problem in spaces of positive constant curvature for the case that all masses are equal and not necessarily constant that any solution for which the point masses move on a circle of not necessarily constant size has to be either a regular cocircular homographic orbit in flat space, or a regular polygonal rotopulsator in curved space, under the constraint that the minimal distance between point masses attains its minimum in finite time. Additionally, we prove that the same holds true if we add an extra mass at the center of that circle and find an explicit formula for the mass of each point particle in terms of the radius of the circle. Finally, we prove that for each order of the masses there is at most one cocircular homographic orbit for the case that the masses need not be constant.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abraham, R., Marsden, J.: Foundations of Mechanics. Addison-Wesley Publishing Co., Reading, MA (1978)

    Google Scholar 

  2. Albouy, A., Cabral, H.E., Santos, A.A.: Some problems on the classical n-body problem. Celest. Mech. Dyn. Astron. 133, 369–375 (2012)

    Google Scholar 

  3. Arribas, M., Elipe, A., Kalvouridis, T., Palacios, M.: Homographic solutions in the planar \(n + 1\)-body problem with quasi-homogeneous potentials. Celest. Mech. Dyn. Astron. 99(1), 1–12 (2007)

    Google Scholar 

  4. Corbera, M., Llibre, J., Pérez-Chavela, E.: Equilibrium points and central configurations for the Lennard-Jones 2- and 3-body problems. Celest. Mech. Dyn. Astron. 89(3), 235–266 (2004)

    Google Scholar 

  5. Cors, J.M., Hall, G.R., Roberts, G.E.: Uniqueness results for co-circular central configurations for power-law potentials. Physica D 280–281, 44–47 (2014)

    Google Scholar 

  6. Craig, S., Diacu, F., Lacomba, E.A., Pérez-Chavela, E.: On the anisotropic Manev problem. J. Math. Phys. 40, 1–17 (1999)

    Google Scholar 

  7. Delgado, J., Diacu, F.N., Lacomba, E.A., Mingarelli, A., Mioc, V., Perez, E., Stoica, C.: The global flow of the Manev problem. J. Math. Phys. 37(6), 2748–2761 (1996)

    Google Scholar 

  8. Diacu, F.: Near-collision dynamics for particle systems with quasihomogeneous potentials. J. Differ. Equ. 128, 58–77 (1996)

    Google Scholar 

  9. Diacu, F.: On the singularities of the curved \(n\)-body problem. Trans. Am. Math. Soc. 363, 2249–2264 (2011)

    Google Scholar 

  10. Diacu, F.: Polygonal homographic orbits of the curved \(n\)-body problem. Trans. Am. Math. Soc. 364(5), 2783–2802 (2012)

    Google Scholar 

  11. Diacu, F.: Relative equilibria in the 3-dimensional curved n-body problem. Memoirs Am. Math. Soc. 228, 1071 (2013)

    Google Scholar 

  12. Diacu, F.: Relative Equilibria of the Curved \(N\)-Body Problem, Atlantis Studies in Dynamical Systems, vol. 1. Atlantis Press, Amsterdam (2012)

    Google Scholar 

  13. Diacu, F.: The non-existence of centre-of-mass and linear-momentum integrals in the curved \(n\)-body problem. arXiv:1202.4739

  14. Diacu, F.: The curved N-body problem: risks and rewards. Math. Intell. 35(3), 24–33 (2013)

    Google Scholar 

  15. Diacu, F., Kordlou, S.: Rotopulsators of the curved N-body problem. J. Differ. Equ. 255, 2709–2750 (2013)

    Google Scholar 

  16. Diacu, F., Pérez-Chavela, E., Santoprete, M.: The n-body problem in spaces of constant curvature. arXiv:0807.1747

  17. Diacu, F., Pérez-Chavela, E., Santoprete, M.: The n-body problem in spaces of constant curvature. Part I: relative equilibria. J. Nonlinear Sci. 22(2), 247–266 (2012)

    Google Scholar 

  18. Diacu, F., Pérez-Chavela, E., Santoprete, M.: The n-body problem in spaces of constant curvature. Part II: singularities. J. Nonlinear Sci. 22(2), 267–275 (2012)

    Google Scholar 

  19. Diacu, F., Pérez-Chavela, E., Santoprete, M.: Central configurations and total collisions for quasihomogeneous \(n\)-body problems. Nonlinear Anal. 65, 1425–1439 (2006)

    Google Scholar 

  20. Diacu, F., Popa, S.: All the Lagrangian relative equilibria of the curved 3-body problem have equal masses. J. Math. Phys. 55, 112701 (2014)

    Google Scholar 

  21. Diacu, F., Thorn, B.: Rectangular orbits of the curved 4-body problem. Proc. Am. Math. Soc. 143, 1583–1593 (2015)

    Google Scholar 

  22. Jones, R.: Central configurations with a quasihomogeneous potential function. J. Math. Phys. 49, 052901 (2008)

    Google Scholar 

  23. Kalvouridis, T.J.: A planar case of the n+1-body problem: the ring problem. Astrophys. Space Sci. 260, 309325 (1999)

    Google Scholar 

  24. Maxwell, J.C.: On the Stability of Motions of Saturns Rings. Macmillan and Cia, Cambridge (1859)

    Google Scholar 

  25. Mioc, V., Stavinschi, M.: On the Schwarzschild-type polygonal (n + 1)-body problem and on the associated restricted problem. Balt. Astron. 7, 637–651 (1998)

    Google Scholar 

  26. Mioc, V., Stavinschi, M.: On Maxwells (n+1)-body problem in the manev-type field and on the associated restricted problem. Phys. Scr. 60, 483–490 (1999)

    Google Scholar 

  27. Muzzio, J.C., Plastino, A.R.: On the use and abuse of Newton’s second law for variable mass problems. Celest. Mech. Dyn. Astron. 53, 227–232 (1992)

    Google Scholar 

  28. Paraschiv, V.: Central configurations and homographic solutions for the quasihomogeneous N-body problem. J. Math. Phys. 53, 122902 (2012)

    Google Scholar 

  29. Pérez-Chavela, E., Saari, D., Susin, A., Yan, Z.: Central configurations in the charged three body problem. Contemp. Math. 198, 137–155 (1996)

    Google Scholar 

  30. Pérez-Chavela, E., Sánchez-Cerritos, J.M.: On the non-existence of hyperbolic polygonal relative equilibria for the negative curved \(n\)-body problem with equal masses. arXiv:1612.09270v1

  31. Tibboel, P.: Polygonal homographic orbits in spaces of constant curvature. Proc. Am. Math. Soc. 141, 1465–1471 (2013)

    Google Scholar 

  32. Tibboel, P.: Existence of a class of rotopulsators. J. Math. Anal. Appl. 404, 185–191 (2013)

    Google Scholar 

  33. Tibboel, P.: Existence of a lower bound for the distance between point masses of relative equilibria in spaces of constant curvature. J. Math. Anal. Appl. 416, 205–211 (2014)

    Google Scholar 

  34. Tibboel, P.: Existence of a lower bound for the distance between point masses of relative equilibria for generalised quasi-homogeneous n-body problems and the curved n-body problem. J. Math. Phys. 56, 032901 (2015)

    Google Scholar 

  35. Tibboel, P.: Finiteness of polygonal relative equilibria for generalised quasi-homogeneous \(n\)-body problems and \(n\)-body problems in spaces of constant curvature. J. Math. Anal. Appl. 441, 183–193 (2016)

    Google Scholar 

  36. Tibboel, P.: Polygonal rotopulsators of the curved n-body problem. J. Math. Phys. 59, 022901 (2018)

    Google Scholar 

  37. Zhu, S.: Eulerian relative equilibria of the curved \(3\)-body problems in \({\mathbf{S}}^2\). Proc. Am. Math. Soc. 142, 2837–2848 (2014)

    Google Scholar 

  38. Zhao, S., Zhu, S.: Three-dimensional central configurations in \({\mathbb{H}}^{3}\) and \({\mathbb{S}}^{3}\). arXiv:1605.08730

Download references

Acknowledgements

The author would like to thank Dr. Alejandro Vidal-López and Dr. Thijs Kouwenhoven of Xi’an Jiaotong-Liverpool University for fruitful discussions that helped the author to double-check the results of this paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Pieter Tibboel.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tibboel, P. Circular Non-collision Orbits for a Large Class of n-Body Problems. J Dyn Diff Equat 32, 205–217 (2020). https://doi.org/10.1007/s10884-018-9714-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-018-9714-7

Keywords

Navigation