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Lobachevskii Journal of Mathematics

, Volume 31, Issue 2, pp 157–173 | Cite as

Group analysis of non-autonomous linear Hamiltonians through differential Galois theory

  • D. Blázquez-Sanz
  • S. A. Carrillo Torres
Article

Abstract

In this paper we introduce a notion of integrability in the non autonomous sense. For the cases of 1 + 1/2 degrees of freedom and quadratic homogeneous Hamiltonians of 2 + 1/2 degrees of freedom we prove that this notion is equivalent to the classical complete integrability of the system in the extended phase space. For the case of quadratic homogeneous Hamiltonians of 2 + 1/2 degrees of freedom we also give a reciprocal of the Morales-Ramis result. We classify those systems by terms of symplectic change of frames involving algebraic functions of time, and give their canonical forms.

Key words and phrases

Hamiltonian Systems Integrability Differential Galois Theory 

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References

  1. 1.
    P. B. Acosta-Humanez, Nonautonomous Hamiltonian systems and Morales-Ramis Theory. I. The case \( \ddot x \) = f(x, t), SIAMJ. Appl. Dyn. Syst. 8(1), 279 (2009).MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    V. I. Arnold, “MathematicalMethods of Classical Mechanics,” Graduate Texts in Mathematics (second edition) (Springer Verlag, 1989), no. 60.Google Scholar
  3. 3.
    D. Blázquez-Sanz, Differential Galois Theory and Lie-Vessiot Systems, (VDM Verlag, 2008).Google Scholar
  4. 4.
    G. Casale, J. Roques, Dynamics of Rational Symplectic Mappings and Difference Galois Theory. Int. Math. Res. Not. IMRN 2008, Art. ID rnn 103, 23 pp.Google Scholar
  5. 5.
    R. C. Churchill, D. L. Rod, and M. F. Singer, M, Group-Theoretic Obstructions to Integrability, Ergodic Theory Dynam. Systems 15(1), 15 (1995).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    J. E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics (Springer Verlag, 1975).Google Scholar
  7. 7.
    I. Kaplansky, An introduction to differential algebra (Hermann, Parias, 1957).MATHGoogle Scholar
  8. 8.
    E. Kolchin, Differential Algebra and Algebraic Groups (Academic Press, New York, 1973).MATHGoogle Scholar
  9. 9.
    A. Maciejewski, M. Przybylska, and H. Yoshida, Necessary conditions for super-integrability of Hamiltonian systems, Phys. Lett. A 372(34), 5581 (2008).CrossRefMathSciNetGoogle Scholar
  10. 10.
    J. J. Morales-Ruiz, Differential Galois Theory and Non-integrability of Hamiltonian Systems (Birskhaüser, 1999).Google Scholar
  11. 11.
    J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of hamiltonian systems I, Methods and Applications of Analysis 8, 39 (2001).Google Scholar
  12. 12.
    J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of hamiltonian systems II, Methods and Applications of Analysis 8, 97 (2001).MATHGoogle Scholar
  13. 13.
    J. J. Morales-Ruiz, C. Simó, and S. Simon, Algebraic proof of the non-integrability of Hill’s problem, Ergodic Theory Dynam. Systems 25(4), 1237 (2005).MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    J. J. Morales-Ruiz and S. Simon, On the meromorphic non-integrability of some N-body problems, Discrete Contin. Dyn. Syst. 24(4), 1225 (2009)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    J. J. Morales-Ruiz, J. P. Ramis, and C. Simó, Integrability of hamiltonian systems abd differential Galois groups of higher order variational equations, Ann. Sci. École Norm. Sup. (4) 40(6), 845 (2007).MATHGoogle Scholar
  16. 16.
    K. Nakagawa, H. Yoshida, A list of all integrable two-dimensional homogeneous polynomial potentials with a polynomial integral of order at most four in the momenta, J. Phys. A 34(41), 8611 (2001).MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    J. Williamson, On the Algebraic Problem Concerning the Normal Forms of Linear Dynamical Systems, Amer. J.Math. 58(1), 141 (1936).CrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Instituto de Matemáticas y sus Aplicaciones (IMA)Universidad Sergio ArboledaBogotáColombia
  2. 2.Facultad de CienciasDepartamento de Matemáticas Universidad Nacional de ColombiaBogotáColombia

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