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Transcendental Case in Stability Problem of Hamiltonian System with Two Degrees of Freedom in Presence of First Order Resonance

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Abstract

We study Lyapunov stability of an equilibrium for Hamiltonian system with two degrees of freedom. We suppose that the matrix of the linearised system has one pair of zero eigenvalues and its Jordan normal form is not diagonal. We consider the so-called transcendental case when usual technique, based on normalization of the Hamiltonian up to a finite order terms does not give an answer to the stability question. We obtain conditions under which the transcendental case takes place and show that the equilibrium is unstable in the transcendental case. We apply the above results to investigate the stability of conic precession of a symmetric satellite in a circular orbit.

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References

  1. Markeev, A.P.: Libration Points in Celestial Mechanics and Space Dynamics. Nauka, Moscow (1978, in Russian)

  2. Markeev A.P.: The problem of the stability of the equilibrium position of a Hamiltonian system at 3:1 resonance. J. Appl. Math. Mech. 65(4), 639–645 (2001)

    Article  MathSciNet  Google Scholar 

  3. Cabral H.E., Meyer K.R.: Stability of equilibria and fixed points of conservative systems. Nonlinearity 12(5), 1351–1362 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Sokolsky A.G.: On the stability of an autonomous Hamiltonian system with two degrees of freedom under first-order resonance. J. Appl. Math. Mech. 41, 20–28 (1977)

    Article  MathSciNet  Google Scholar 

  5. Dos Santos F., Vidal C.: Stability of equilibrium solutions of Hamiltonian systems under the presence of a single resonance in the non-diagonalizable case. Regul. Chaotic Dyn. 13(3), 166–177 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Willianson J.: On the algebraic problem concerning the normal forms of linear dynamical systems. Am. J. Math. 58, 141–163 (1936)

    Article  Google Scholar 

  7. Bulgakov B.V.: On normal coordinates. Prikl. Mat. Mekh. 10(2), 273–294 (1946)

    MathSciNet  MATH  Google Scholar 

  8. Galin, D.M.: Versal deformations of linear Hamiltonian systems. In: Tr. Semin. Im Petrovskogo, vol. 1, pp. 63–74, 1975. Engl. transl.: Transl., Ser. 2, Am. Math. Soc. 118, 1–12 (1982)

  9. Maciejewski, A.J.: Ruch obrotowy bryly sztywnej umieszczonej w punkcie libracyjnym ograniczonego zagadnienia trzech cial. Ph.D. thesis, Torun (1987)

  10. Markeev, A.P.: Stability of plane oscillations and rotations of a satellite on a circular orbit. Kosm. Issled. 13, 322–336 (May 1975). Engl. translation in Cosmic Research, 13(3) 285–298 (Nov. 1975)

  11. Kondurar V.T.: Particular solutions of the general problem of the translational-rotational motion of a spheroid attracted by a sphere. Sov. Astron. AJ 3, 863–875 (1959)

    MathSciNet  Google Scholar 

  12. Duboshin, G.N.: Rotational motion of artificial celestial bodies. Byull. Inst. Teor. Astron. Akad. Nauk SSSR 7, 511–520 (1960, in Russian)

    Google Scholar 

  13. Beletskii, V.V.: Motion of a Satellite About Its Center of Mass in the Gravitational Field. University Press, Moscow (1975, in Russian)

  14. Likins P.W.: Stability of a symmetrical satellite in attitudes fixed in an orbiting reference frame. J. Astronaut. Sci. 12, 18–24 (1965)

    Google Scholar 

  15. Markeev A.P.: Resonance effects and stability of stationary motion of a satellite. Kosm. Issled. 5(3), 365–375 (1967)

    Google Scholar 

  16. Markeev A.P.: Stability of a canonical system with two degrees of freedom in the presence of resonance. J. Appl. Math. Mech. 32, 766–772 (1968)

    Article  MathSciNet  Google Scholar 

  17. Sokolsky A.G.: On problem of stability of regular precessions of symmetric satellite. Kosm. Issled. 18(5), 698–706 (1980)

    Google Scholar 

  18. Markeev A.P., Skolskii A.G., Chekhovskaia T.N.: Stability of the conical precession of a dynamically symmetric rigid body. Sov. Astron. Lett. 3, 178–180 (1977)

    Google Scholar 

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Authors and Affiliations

  1. Department of Theoretical Mechanics, Faculty of Applied Mathematics, Moscow Aviation Institute, 4 Volokolamskoe Shosse, Moscow, 125871, Russia

    Boris S. Bardin

  2. Kepler Institute of Astronomy, University of Zielona Góra, Licealna 9, 65-417, Zielona Góra, Poland

    Andrzej J. Maciejewski

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  1. Boris S. Bardin
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  2. Andrzej J. Maciejewski
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Correspondence to Boris S. Bardin.

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Bardin, B.S., Maciejewski, A.J. Transcendental Case in Stability Problem of Hamiltonian System with Two Degrees of Freedom in Presence of First Order Resonance. Qual. Theory Dyn. Syst. 12, 207–216 (2013). https://doi.org/10.1007/s12346-012-0077-x

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  • DOI: https://doi.org/10.1007/s12346-012-0077-x

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