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.
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References
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).
V. I. Arnold, “MathematicalMethods of Classical Mechanics,” Graduate Texts in Mathematics (second edition) (Springer Verlag, 1989), no. 60.
D. Blázquez-Sanz, Differential Galois Theory and Lie-Vessiot Systems, (VDM Verlag, 2008).
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.
R. C. Churchill, D. L. Rod, and M. F. Singer, M, Group-Theoretic Obstructions to Integrability, Ergodic Theory Dynam. Systems 15(1), 15 (1995).
J. E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics (Springer Verlag, 1975).
I. Kaplansky, An introduction to differential algebra (Hermann, Parias, 1957).
E. Kolchin, Differential Algebra and Algebraic Groups (Academic Press, New York, 1973).
A. Maciejewski, M. Przybylska, and H. Yoshida, Necessary conditions for super-integrability of Hamiltonian systems, Phys. Lett. A 372(34), 5581 (2008).
J. J. Morales-Ruiz, Differential Galois Theory and Non-integrability of Hamiltonian Systems (Birskhaüser, 1999).
J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of hamiltonian systems I, Methods and Applications of Analysis 8, 39 (2001).
J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of hamiltonian systems II, Methods and Applications of Analysis 8, 97 (2001).
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).
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)
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).
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).
J. Williamson, On the Algebraic Problem Concerning the Normal Forms of Linear Dynamical Systems, Amer. J.Math. 58(1), 141 (1936).
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Submitted by N.H. Ibragimov
An erratum to this article is available at http://dx.doi.org/10.1134/S1995080211030073.
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Blázquez-Sanz, D., Carrillo Torres, S.A. Group analysis of non-autonomous linear Hamiltonians through differential Galois theory. Lobachevskii J Math 31, 157–173 (2010). https://doi.org/10.1134/S1995080210020071
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DOI: https://doi.org/10.1134/S1995080210020071