On the integrability of 2D Hamiltonian systems with variable Gaussian curvature

Original Paper


In this work, we consider the integrability of a general 2D motion of a particle on a surface with variable Gaussian curvature under the influence of conservative potential forces. Although this system has a kinetic energy relying on the coordinates, it remains homogeneous. The homogeneity of the system generally enables us to find a particular solution that can be utilized to derive the necessary conditions for the integrability by studying the properties of the differential Galois group of the normal variational equations along this particular solution. We present a new theory that can be applied to determine the necessary conditions for the integrability of Hamiltonian systems having a variable Gaussian curvature.


Liouville integrability Differential Galois theory Systems in polar coordinates 



We would like to express our thanks to the reviewers for their useful comments which enabled us to improve the presentation of this article.


  1. 1.
    Babelon, O., Bernard, D., Talon, M.: Introduction to Classical Integrable Systems. Cambridge University Press, Cambridge (2003)CrossRefMATHGoogle Scholar
  2. 2.
    Tabor, M.: Chaos and Integrability in Nonlinear Dynamics. Wiley, NewYork (1988)Google Scholar
  3. 3.
    Elmandouh, A.A.: New integrable problems in rigid body dynamics with quartic integrals. Acta Mech. 226(8), 2461–2472 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Elmandouh, A.A.: New integrable problems in the dynamics of particle and rigid body. Acta Mech. 226(11), 3749–3762 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Elmandouh, A.A.: On the integrability of the motion of 3D-swinging Atwood machine and related problems. Phys. Lett. A 380(9), 989–991 (2016)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Yehia, H.M.: Atlas of two-dimensional irreversible conservative Lagrangian mechanical systems with a second quadratic integral. J. Math. Phys. 48(8), 082902 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Yehia, H.M., Elmandouh, A.A.: A new integrable problem with a quartic integral in the dynamics of a rigid body. J. Phys. A Math. Theor. 46(14), 142001 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Karlovini, M., Pucacco, G., Rosquist, K., Samuelsson, L.: A unified treatment of quartic invariants at fixed and arbitrary energy. J. Math. Phys. 43(8), 4041–4059 (2002)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hu, Z., Aldazharova, M., Aldibekov, T.M., Romanovski, V.G.: Integrability of 3-dim polynomial systems with three invariant planes. Nonlinear Dyn. 74(4), 1077–1092 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Llibre, J., Ramirez, R., Sadovskaia, N.: Integrability of the constrained rigid body. Nonlinear Dyn. 73(4), 2273–2290 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Llibre, J., Oliveira, R.D., Valls, C.: On the integrability and the zero-Hopf bifurcation of a Chen–Wang differential system. Nonlinear Dyn. 80(1–2), 353–361 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Bao, J., Yang, Q.: Darboux integrability of the stretch-twist-fold flow. Nonlinear Dyn. 76(1), 797–807 (2014)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Lima, M.F., Llibre, J., Valls, C.: Integrability of the Rucklidge system. Nonlinear Dyn. 77(4), 1441–1453 (2014)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Bountis, T., Segur, H., Vivaldi, F.: Integrable Hamiltonian systems and the Painlevé property. Phys. Rev. A 25(3), 1257 (1982)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ziglin, S.L.V.: Branching of solutions and nonexistence of first integrals in Hamiltonian mechanics. I. Funct. Anal. Appl. 16(3), 181–189 (1982)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Morales-Ruiz, J.J.: Differential Galois Theory and Non-integrability of Hamiltonian Systems. Prog. Math. Birkhauser Verlag, Basel (1999)CrossRefMATHGoogle Scholar
  17. 17.
    Morales-Ruiz, J.J., Ramis, J.P.: 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. 18.
    Gu, C., Hu, A., Zhou, Z.: Darboux Transformations in Integrable Systems: Theory and Their Applications to Geometry. Springer, Berlin (2006)MATHGoogle Scholar
  19. 19.
    Wazwaz, A.M., Xu, G.Q.: An extended modified KdV equation and its Painlevé integrability. Nonlinear Dyn. 86(3), 1455–1460 (2016)CrossRefMATHGoogle Scholar
  20. 20.
    Wazwaz, A.M., El-Tantawy, S.A.: A new integrable (3+1)-dimensional KdV-like model with its multiple-soliton solutions. Nonlinear Dyn. 83(3), 1529–1534 (2016)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Birkhoff, G.D.: Dynamical Systems. American Mathematical Society, New York (1927)CrossRefMATHGoogle Scholar
  22. 22.
    Yoshida, H.: A criterion for the non-existence of an additional integral in Hamiltonian systems with a homogeneous potential. Physica D 29(1), 128–142 (1987)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Pogorelov, A.V.: Differential Geometry. Noordhoff, Geoningen (1954)MATHGoogle Scholar
  24. 24.
    Szumiński, W., Maciejewski, A.J., Przybylska, M.: Note on integrability of certain homogeneous Hamiltonian systems. Phys. Lett. A 379(45), 2970–2976 (2015)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Maciejewski, A.J., Szumiński, W., Przybylska, M.: Note on integrability of certain homogeneous Hamiltonian systems in 2D constant curvature spaces. Phys. Lett. A 381(7), 725–732 (2017)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Casale, G., Duval, G., Maciejewski, A.J., Przybylska, M.: Integrability of Hamiltonian systems with homogenous potential of degree zero. Phys. Lett. A 374, 448–452 (2010)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Raňada, M.F., Santander, M.: Superintegrable systems on the two-dimensional sphere \(S^2\) and hyperbolic plane \(H^2\). J. Math. Phys. 40, 5026–5057 (1999)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Combot, T.: A note on algebraic potentials and Morales–Ramis theory. Celest. Mech. Dyn. Astron. 115(4), 397–404 (2013)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Cambridge University Press, Cambridge (1996)CrossRefMATHGoogle Scholar
  30. 30.
    Kimura, T.: On Riemann’s equations which are solvable by quadratures. Funkc. Ekvacioj 12, 269–281 (1969)MathSciNetMATHGoogle Scholar
  31. 31.
    Atwood, G.: A Treatise on the Rectilinear Motion and Rotation of Bodies. Cambridge University Press, Cambridge (1784)Google Scholar
  32. 32.
    Morales-Ruiz, J.J., Ramis, J.P.: A note on the non-integrability of some Hamiltonian systems with a homogeneous potential. Methods Appl. Anal. 8, 113120 (2001)MathSciNetMATHGoogle Scholar
  33. 33.
    Casasayas, J., Nunes, A., Tufillaro, N.: Swinging Atwood’s machine: Integrability and dynamics. J. Phys. France 51, 16931702 (1990)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Tufillaro, N.: Integrable motion of a swinging Atwood’s machine. Am. J. Phys. 54(2), 142–143 (1986)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, Faculty of ScienceKing Faisal UniversityAl-AhsaaSaudi Arabia
  2. 2.Department of Mathematics, Faculty of ScienceMansoura UniversityMansouraEgypt

Personalised recommendations