Celestial Mechanics and Dynamical Astronomy

, Volume 112, Issue 2, pp 149–167 | Cite as

Non-integrability of first order resonances in Hamiltonian systems in three degrees of freedom

Original Article


The normal forms of the Hamiltonian 1:2:ω resonances to degree three for ω = 1, 3, 4 are studied for integrability. We prove that these systems are non-integrable except for the discrete values of the parameters which are well known. We use the Ziglin–Morales–Ramis method based on the differential Galois theory.


Resonance Non-integrability Morales–Ramis theory 


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  1. Arnold, V., Kozlov, V., Neishtadt, A.: Mathematical Aspects of Classical and Celestial Mechanics. In: Dynamical Systems III. Springer, New York (2006)Google Scholar
  2. Baider A., Churchill R., Rod D.: Monodromy and non-integrability in complex Hamiltonian systems. J. Dyn. Differ. Equ. 2, 451–481 (1990)MathSciNetMATHCrossRefGoogle Scholar
  3. Duistermaat J.: Non-integrability of the 1:1:2 resonance. Ergod. Theory Dyn. Syst. 4, 553–568 (1984)MathSciNetMATHCrossRefGoogle Scholar
  4. Ford, J.: The statistical mechanics of classical analytic dynamics. In: Cohen, E.D.G. (ed.) Fundamental Problems in Statistical Mechanics, vol. 3. (1975)Google Scholar
  5. Haller G., Wiggins S.: Geometry and chaos near resonant equilibria of 3-DOF Hamiltonian systems. Phys. D 90, 319–365 (1996)MathSciNetMATHCrossRefGoogle Scholar
  6. Horozov E.: On the non-integrability of the Gross-Neveu models. Ann. Phys. 174, 430–441 (1987)MathSciNetADSMATHCrossRefGoogle Scholar
  7. Hoveijn I., Verhulst F.: Chaos in the 1:2:3 Hamiltonian normal form. Phys. D 44, 397–406 (1990)MathSciNetMATHCrossRefGoogle Scholar
  8. Il’yashenko, Y., Yakovenko, S.: Lectures on Analytic Theory of Ordinary Differential Equations, Graduate Studies in Mathematics, vol. 86. AMS, Providence (2008)Google Scholar
  9. Ince E.: Ordinary Differential Equations. Dover, New York (1956)Google Scholar
  10. Irigoyen M., Simó C.: Non-integrability of the J2 problem. Celest. Mech. Dyn. Astron. 55, 281–287 (1993)ADSMATHCrossRefGoogle Scholar
  11. Iwasaki, K., Kimura, H., Shimomura, S., Yoshida, M.: From Gauss to Painlevé : A Modern Theory of Special Functions, Aspects of Mathematics, vol. 16, Vieweg (1991)Google Scholar
  12. Kimura T.: On Riemann’s equations which are solvable by quadratures. Funkcialaj Ekvaciaj 12, 269–281 (1969)MATHGoogle Scholar
  13. Li W., Shi S.: Non-integrability of Hénon-Heiles system. Celest. Mech. Dyn. Astron. 109, 1–12 (2010). doi: 10.1007/s10569-010-9315-1 MathSciNetADSCrossRefGoogle Scholar
  14. Maciejewski A., Przybilska M.: Non-integrability of the three-body problem. Celest. Mech. Dyn. Astron. 110, 17–30 (2011). doi: 10.1007/s10569-010-9333-z ADSCrossRefGoogle Scholar
  15. Martinet L., Magnenat P., Verhulst F.: On the number of isolating integrals in resonant systems with 3 degrees of freedom. Celest. Mech. 25, 93–99 (1981)MathSciNetADSMATHCrossRefGoogle Scholar
  16. Meyer K., Hall G.: Introduction to Hamiltonian dynamical systems and the N-body problem. Springer, New York (1992)MATHGoogle Scholar
  17. Morales Ruiz, J.: Differential Galois Theory and Non integrability of Hamiltonian Systems, Progress in Mathematics, vol. 179. Birkhäuser (1999)Google Scholar
  18. Morales-Ruiz J., Ramis J.-P., Simó C.: Integrability of Hamiltonian systems and differential Galois groups of higher variational equations. Ann. Sci. Ecol. Norm. Sup. 40, 845–884 (2007)MATHGoogle Scholar
  19. Morales-Ruiz, J., Ramis, J.-P.: Integrability of Dynamical systems Through Differential Galois Theory: Practical Guide, Contemporary Mathematics, vol. 509. (2010)Google Scholar
  20. Morales J.J., Simó C., Simón S.: Algebraic proof of the non-integrability of Hill’s problem. Ergod. Theory Dyn. Syst. 25, 1237–1256 (2005)MATHCrossRefGoogle Scholar
  21. Singer M., van der Put M.: Galois Theory of Linear Differential Equations, Grundlehren der Mathematischen Wissenschaften. Vol. 328. Springer, Berlin (2003)Google Scholar
  22. Tsygvintsev A.: The meromorphic non-integrability of the three-body problem. J. Reine Angew. Math. 537, 127–149 (2001)MathSciNetMATHCrossRefGoogle Scholar
  23. Tsygvintsev A.: On some exceptional cases in the integrability of the three-body problem. Celest. Mech. Dyn. Astron. 99, 23–29 (2007)MathSciNetADSMATHCrossRefGoogle Scholar
  24. van der Aa E.: First order resonances in three degrees of freedom systems. Celest. Mech. 31, 163–191 (1983)ADSMATHCrossRefGoogle Scholar
  25. van der Aa, E., Sanders, J.: The 1:2:1 Resonance, its periodic orbits and integrals. In: Lecture Notes in Mathematics, vol. 711. Springer, New York (1979)Google Scholar
  26. van der Aa E., Verhulst F.: Asymptotic integrability and periodic solutions of a Hamiltonian system in 1:2:2 resonance. SIAM J. Math. Anal. 15, 890–911 (1984)MathSciNetMATHCrossRefGoogle Scholar
  27. Verhulst F.: Symmetry and integrability in Hamiltonian normal forms. In: Bambusi, D., Gaeta, G. (eds) Symmetry and Perturbation Theory, pp. 245–284. Quadern GNFM, Farenze (1998)Google Scholar
  28. Verhulst F., Hoveijn I.: Integrability and chaos in Hamiltonian normal forms. In: Broer, H., Takens, F. (eds) Geometry and Analysis in Nonlinear Dynamics, Pitman, London (1992)Google Scholar
  29. Wasov W.: Asymptotic Expansions for Ordinary Differential Equations. Dover, New York (1965)Google Scholar
  30. Yoshida H.: On the class of variational equations transformable to the Gauss hypergeometric equation. Celest. Mech. Dyn. Astron. 53, 145–150 (1992)ADSMATHCrossRefGoogle Scholar
  31. Ziglin S.: Branching of solutions and non-existence of first integrals in Hamiltonian mechanics. I. Funct. Anal. Appl. 16, 30–41 (1982)MathSciNetGoogle Scholar
  32. Ziglin S.: Branching of solutions and non-existence of first integrals in Hamiltonian mechanics. II. Funct. Anal. Appl. 17, 8–23 (1983)MathSciNetCrossRefGoogle Scholar
  33. Zoladek H.: The Monodromy Group (Monografie Matematyczne). Birkhäuser, Basel (2006)Google Scholar

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© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Mathematics and InformaticsSofia UniversitySofiaBulgaria

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