Regular and Chaotic Dynamics

, Volume 23, Issue 7–8, pp 961–973 | Cite as

On the Sectional Curvature Along Central Configurations

  • Connor JackmanEmail author
  • Josué Meléndez


In this paper we characterize planar central configurations in terms of a sectional curvature value of the Jacobi–Maupertuis metric. This characterization works for the N-body problem with general masses and any 1/rα potential with α > 0. We also obtain dynamical consequences of these curvature values for relative equilibrium solutions. These curvature methods work well for strong forces (α ≥ 2).


instability homographic solutions central configurations Jacobi–Maupertuis metric 

MSC2010 numbers

70F10 37N05 70G45 


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  1. 1.
    Albouy, A., Cabral, H.E., and Santos, A.A., Some Open Problems on the Classical N-Body Problem, Celestial Mech. Dynam. Astronom., 2012, vol. 113, no. 4, pp. 369–375.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Arnol’d, V. I., Mathematical Methods of Classical Mechanics, 2nd ed., Grad. Texts in Math., vol. 60, New York: Springer, 1989.Google Scholar
  3. 3.
    Barbosu, M. and Elmabsout, B., Courbures de Riemann dans le problème plan des trois corps: Stabilité, C. R. Acad. Sci. Paris. Sér. 2b, 1999, vol. 327, pp. 959–962.zbMATHGoogle Scholar
  4. 4.
    Barutello, V., Jadanza, R. D., and Portaluri, A., Morse Index and Linear Stability of the Lagrangian Circular Orbit in a Three-Body-Type Problem via Index Theory, Arch. Ration. Mech. Anal., 2016, vol. 219, no. 1, pp. 387–444.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chenciner, A., Collisions totales, mouvements complètement paraboliques et réduction des homothéties dans le problème des n corps, Regul. Chaotic Dyn., 1998, vol. 3, no. 3, pp. 93–106.zbMATHGoogle Scholar
  6. 6.
    Fujiwara, T., Fukuda, H., Ozaki, H., and Taniguchi, T., Saari’s Homographic Conjecture for General Masses in Planar Three-Body Problem under Newton Potential and a Strong Force Potential, J. Phys. A, 2015, vol. 48, no. 26, 265205, 17 pp.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hu, X. and Sun, Sh., Morse Index and Stability of Elliptic Lagrangian Solutions in the Planar Three-Body Problem, Adv. Math., 2010, vol. 223, no. 1, pp. 98–119.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hu, X. and Yu, G., An Index Theory for Zero Energy Solutions of the Planar Anisotropic Kepler Problem, Comm. Math. Phys., 2018, vol. 361, no. 2, pp. 709–736.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jackman, C. and Meléndez, J., Hyperbolic Shirts Fit a 4-Body Problem, J. Geom. Phys., 2018, vol. 123, pp. 173–183.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jackman, C. and Montgomery, R., No Hyperbolic Pants for the 4-Body Problem with Strong Potential, Pacific J. Math., 2016, vol. 280, no. 2, pp. 401–410.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kobayashi, Sh., Hyperbolic Complex Spaces, Grundlehren Math. Wiss., vol. 318, Berlin: Springer, 1998.Google Scholar
  12. 12.
    Lee, J.M., Riemannian Manifolds: An Introduction to Curvature, Grad. Texts in Math., vol. 176, New York: Springer, 1997.Google Scholar
  13. 13.
    Martínez, R., Samà, A., and Simó, C., Stability of Homographic Solutions of the Planar Three-Body Problem with Homogeneous Potentials, in Proc. of the Internat. Conf. on Differential Equations (EQUADIFF’2003), F. Dumortier, H. Broer, J. Mawhin, A. Van der bauwhede, S. Verduyn Lunel (Eds.), Hackensack,N.J.: World Sci., 2005, pp. 1005–1010.Google Scholar
  14. 14.
  15. 15.
    Moeckel, R., A Computer Assisted Proof of Saari’s Conjecture for the Three-Body Problem in Rd, Discrete Contin. Dyn. Syst. Ser. S, 2008, vol. 1, no. 4, pp. 631–646.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Montgomery, R., The Hyperbolic Plane, Three-Body Problems, andMnëv’s Universality Theorem, Regul. Chaotic Dyn., 2017, vol. 22, no. 6, pp. 688–699.CrossRefzbMATHGoogle Scholar
  17. 17.
    Montgomery, R., Fitting Hyperbolic Pants to a Three-Body Problem, Ergodic Theory Dynam. Systems, 2005, vol. 25, no. 3, pp. 921–947.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Montgomery, R., Blow-Up for Realizing Homotopy Classes in the Three-Body Problem, arXiv:1507.07982 (2015).Google Scholar
  19. 19.
    Pin, O.Ch., Curvature and Mechanics, Advances in Math., 1975, vol. 15, pp. 269–311.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Paternain, G.P., Geodesic Flows, Progr. Math., vol. 180, Boston,Mass.: Birkhüser, 2012.Google Scholar
  21. 21.
    Roberts, G. E., Some Counterexamples to a Generalized Saari’s Conjecture, Trans. Amer. Math. Soc., 2006, vol. 358, no. 1, pp. 251–265.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Roberts, G.E., Linear Stability of the Elliptic Lagrangian Triangle Solutions in the Three-Body Problem, J. Differential Equations, 2002, vol. 182, no. 1, pp. 191–218.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.UC Santa CruzSanta CruzUSA
  2. 2.UAM–Iztapalapa San Rafael Atlixco 186Código PostalMéxico

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