Regular and Chaotic Dynamics

, Volume 18, Issue 5, pp 497–507 | Cite as

Non-integrability of a self-gravitating riemann liquid ellipsoid



We consider the motion of a triaxial Riemann ellipsoid of a homogeneous liquid without angular momentum. We prove that it does not admit an additional first integral which is meromorphic in position, impulsions, and elliptic integrals which appear in the potential. This proves that the system is not integrable in the Liouville sense; we actually show that even its restriction to a fixed energy hypersurface is not integrable.


Morales-Ramis theory elliptic functions monodromy differential Galois theory Riemann surfaces 

MSC2010 numbers



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  1. 1.
    Ayoul, M. and Zung, N.T., Galoisian Obstructions to Non-Hamiltonian Integrability, C. R. Math. Acad. Sci. Paris, 2010, vol. 348, nos. 23–24, pp. 1323–1326.MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bogoyavlensky, O. I., Extended Integrability and Bi-Hamiltonian Systems, Comm. Math. Phys., 1998, vol. 196, no. 1, pp. 19–51.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., The Hamiltonian Dynamics of Self-Gravitating Liquid and Gas Ellipsoids, Regul. Chaotic Dyn., 2009, vol. 14, no. 2, pp. 179–217.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Combot, Th., A Note on Algebraic Potentials and Morales-Ramis Theory, Celestial Mech. Dynam. Astronom., 2013, vol. 115, no. 4, pp. 397–404.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Combot, Th., Non-Integrability of the Equal Mass n-Body Problem with Non-Zero Angular Momentum, Celestial Mech. Dynam. Astronom., 2012, vol. 114, no. 4, pp. 319–340.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Hill, M. J. M., Note on the Motion of a Fluid Ellipsoid under Its Own Attraction, Proc. London Math. Soc., 1891, S1–23, no. 1, p. 88.CrossRefGoogle Scholar
  7. 7.
    Kovacic, J. J., An Algorithm for Solving Second Order Linear Homogeneous Differential Equations, J. Symbolic Comput., 1986, vol. 2, no. 1, pp. 3–43.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Maciejewski, A. J. and Przybylska, M., Non-Integrability of the Generalized Two Fixed Centres Problem, Celestial Mech. Dynam. Astronom., 2004, vol. 89, no. 2, pp. 145–164.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Morales-Ruiz, J. J., Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Progr. Math., vol. 179, Basel: Birkhäuser, 1999.CrossRefMATHGoogle Scholar
  10. 10.
    Morales-Ruiz, J. J. and Ramis, J.-P., Galoisian Obstructions to Integrability of Hamiltonian Systems: 1, 2, Methods Appl. Anal., 2001, vol. 8, no. 1, pp. 33–95, 97–111.MathSciNetMATHGoogle Scholar
  11. 11.
    Rosensteel, G. and Tran, H.Q., Hamiltonian Dynamics of Self-Gravitating Ellipsoids, Astrophys. J., 1991, vol. 366, pp. 30–37.CrossRefGoogle Scholar
  12. 12.
    Shafarevich, I.R., Basic Algebraic Geometry: In 2 Vols., 2nd ed., Berlin: Springer, 1994.CrossRefGoogle Scholar
  13. 13.
    van der Put, M. and Singer, M. F., Galois Theory of Linear Differential Equations, Grundlehren Math. Wiss., vol. 328, Berlin: Springer, 2003.CrossRefMATHGoogle Scholar
  14. 14.
    Ziglin, S. L. On the Absence of an Additional Meromorphic First Integral in the Riemann Problem on the Motion of a Homogeneous Liquid Ellipsoid, Regul. Chaotic Dyn., 2010, vol. 15, nos. 4–5, pp. 630–633.MathSciNetCrossRefMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.IMCCEParisFrance

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