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Hamiltonian geometric connection associated with adiabatically perturbed Hamiltonian systems and the existence of adiabatic invariants

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We study the differential-geometric properties of Hamiltonian connections on symplectic manifolds for adiabatically perturbed Hamiltonian systems. In particular, an associated Hamiltonian connection is constructed on the principal fibration. Its description is given in terms of covariant derivatives and the curvature form of the corresponding connection.

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References

  1. D. Blackmore, Y. A. Prykarpatsky, and R. Samulyak, “The integrability of Lie-invariant geometric objects generated by ideals in the Grassmann algebra,” J. Nonlin. Math. Phys., 5, No. 1, 54–67 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  2. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Wiley, New York (1963, 1969).

    MATH  Google Scholar 

  3. V. I. Arnol’d, “Instability of dynamical systems with many degrees of freedom,” Dokl. Akad. Nauk SSSR, 156, 9–12 (1964).

    MathSciNet  Google Scholar 

  4. L. V. Kantorovich and G. P. Akilov, Functional Analysis [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  5. R. Abraham and J. Marsden, Foundations of Mechanics, Springer, New York (1978).

    MATH  Google Scholar 

  6. V. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Contemporary Geometry [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  7. A. S. Mishchenko and A. T. Fomenko, A Course of Differential Geometry and Topology [in Russian], Moscow University, Moscow (1980).

    Google Scholar 

  8. R. Montgomery, “The connection whose holonomy is the classical adiabatic angles of Hannay and Berry and its generalization to the nonintegrable case,” Commun. Math. Phys., 120, 269–294 (1988).

    Article  MATH  Google Scholar 

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 3, pp. 382–387, March, 2008.

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Prykarpats’kyi, Y.A. Hamiltonian geometric connection associated with adiabatically perturbed Hamiltonian systems and the existence of adiabatic invariants. Ukr Math J 60, 441–448 (2008). https://doi.org/10.1007/s11253-008-0066-z

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  • DOI: https://doi.org/10.1007/s11253-008-0066-z

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