Celestial Mechanics and Dynamical Astronomy

, Volume 114, Issue 4, pp 319–340 | Cite as

Non-integrability of the equal mass n-body problem with non-zero angular momentum

Original Article


We prove an integrability criterion and a partial integrability criterion for homogeneous potentials of degree −1 which are invariant by rotation. We then apply it to the proof of the meromorphic non-integrability of the n-body problem with Newtonian interaction in the plane on a surface of equation (H, C) = (H0, C0) with (H0, C0) ≠ (0, 0) where C is the total angular momentum and H the Hamiltonian, in the case where the n masses are equal. Several other cases in the 3-body problem are also proved to be non integrable in the same way, and some examples displaying partial integrability are provided.


Non-integrability Homogeneous potentials Central configurations Differential Galois theory n-body problem 

Mathematics Subject Classification

37J30 70F15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ayoul M., Zung N.: Galoisian obstructions to non-Hamiltonian integrability. Comptes Rendus Mathematique 348(23), 1323–1326 (2010)MathSciNetCrossRefMATHGoogle Scholar
  2. Bogoyavlensky O.: Extended integrability and bi-Hamiltonian systems. Commun. Math. Phys. 196(1), 19–51 (1998)ADSCrossRefGoogle Scholar
  3. Boucher D.: Sur la non-intégrabilité du probleme plan des trois corps de masses égalesa un le long de la solution de Lagrange. CR Acad. Sci. Paris 331, 391–394 (2000)MathSciNetADSCrossRefMATHGoogle Scholar
  4. Corbera M., Delgado J., Llibre J.: On the existence of central configurations of p nested n-gons. Qualitat. Theory Dyn. Syst. 8, 255–265 (2009). doi:10.1007/s12346-010-0004-y MathSciNetCrossRefGoogle Scholar
  5. Craven B.: Complex symmetric matrices. J. Aust. Math. Soc. 10(3–4), 341–354 (1969)MathSciNetCrossRefMATHGoogle Scholar
  6. Gantmacher, F.: Matrix Theory, vol. 2. New York (1959)Google Scholar
  7. Hietarinta J.: A search for integrable two-dimensional Hamiltonian systems with polynomial potential. Phys. Lett. A 96(6), 273–278 (1983)MathSciNetADSCrossRefGoogle Scholar
  8. Howard J.E., Meiss J.D.: Straight line orbits in Hamiltonian flows. Celest. Mech. Dyn. Astron. 105, 337–352 (2009). doi:10.1007/s10569-009-9231-4,0903.4995 MathSciNetADSCrossRefMATHGoogle Scholar
  9. Kimura T.: On Riemann’s equations which are solvable by quadratures. Funkcial Ekvac 12(269–281), 1970 (1969)Google Scholar
  10. Kovacic J.: An algorithm for solving second order linear homogeneous differential equations. J. Symbol. Comput. 2(1), 3–43 (1986)MathSciNetCrossRefMATHGoogle Scholar
  11. Lee T., Santoprete M.: Central configurations of the five-body problem with equal masses. Celest. Mech. Dyn. Astron. 104(4), 369–381 (2009)MathSciNetADSCrossRefMATHGoogle Scholar
  12. Maciejewski, A., Przybylska, M., Yoshida, H.: Necessary conditions for partial and super-integrability of Hamiltonian systems with homogeneous potentia. Arxiv preprint nlin/0701057 (2007)Google Scholar
  13. Maciejewski A.J., Przybylska M.: Non-integrability of the three-body problem. Celest. Mech. Dyn. Astron. 110(1), 17–30 (2011)MathSciNetADSCrossRefGoogle Scholar
  14. Morales Ruiz, J.: Differential Galois theory and non-integrability of Hamiltonian systems. Progress Math. Boston, vol. 179 (1999)Google Scholar
  15. Morales-Ruiz J., Ramis J.: Galoisian obstructions to integrability of Hamiltonian systems. Methods Appl. Anal. 8(1), 33–96 (2001)MathSciNetMATHGoogle Scholar
  16. Morales-Ruiz J., Ramis J.: Galoisian obstructions to integrability of Hamiltonian systems ii. Methods Appl. Anal. 8(1), 97–112 (2001)MATHGoogle Scholar
  17. Morales-Ruiz J., Ramis J.: A note on the non-integrability of some Hamiltonian systems with a homogeneous potential. Methods Appl. Anal. 8(1), 113–120 (2001)MathSciNetMATHGoogle Scholar
  18. Morales-Ruiz J., Simon S.: On the meromorphic non-integrability of some n-body problems. Discrete Continuous Dyn. Syst. (DCDS-A) 24(4), 1225–1273 (2009)MathSciNetCrossRefMATHGoogle Scholar
  19. Morales-Ruiz J., Ramis J., Simo C.: Integrability of Hamiltonian systems and differential Galois groups of higher variational equations. Annal. Scient. l’Ecole Normale Supérieure 40, 845–884 (2007)MathSciNetCrossRefMATHGoogle Scholar
  20. Pacella F.: Central configurations of the n-body problem via equivariant morse theory. Arch. Ration. Mech. Anal. 97(1), 59–74 (1987)MathSciNetCrossRefMATHGoogle Scholar
  21. Pina E., Lonngi P.: Central configurations for the planar Newtonian four-body problem. Celest. Mech. Dyn. Astron. 108(1), 73–93 (2010)MathSciNetADSCrossRefMATHGoogle Scholar
  22. Ronveaux A.: Heun’s Differential Equations. Oxford University Press, Oxford (1995)MATHGoogle Scholar
  23. Sarlet, W., Cantrijn, F.: Generalizations of Noether’s theorem in classical mechanics. Siam Review, pp. 467–494 (1981)Google Scholar
  24. Tsygvintsev A.: The meromorphic non-integrability of the three-body problem. Journal fuer die reine und angewandte Mathematik (Crelles Journal) 2001(537), 127–149 (2001)MathSciNetGoogle Scholar
  25. Tsygvintsev A.: On some exceptional cases in the integrability of the three-body problem. Celest. Mech. Dyn. Astron. 99(1), 23–29 (2007)MathSciNetADSCrossRefMATHGoogle Scholar
  26. Yoshida H.: Necessary condition for the existence of algebraic first integrals. Celest. Mech. 31(4), 381–399 (1983)ADSCrossRefMATHGoogle Scholar
  27. Yoshida H.: A criterion for the non-existence of an additional integral in Hamiltonian systems with a homogeneous potential. Phys. D Nonlinear Phenom. 29(1–2), 128–142 (1987)ADSCrossRefMATHGoogle Scholar
  28. Ziglin S.: Branching of solutions and nonexistence of first integrals in Hamiltonian mechanics. i. Function. Anal. Appl. 16(3), 181–189 (1982)MathSciNetCrossRefGoogle Scholar
  29. Ziglin S.: Integrals in involution for groups of linear symplectic transformations and natural mechanical systems with homogeneous potential. Function. Anal. Appl. 34(3), 179–187 (2000)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.IMCCEParisFrance

Personalised recommendations