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Hamiltonian structure, equilibria, and stability for an axisymmetric gyrostat motion in the presence of gravity and magnetic fields

  • A. A. ElmandouhEmail author
  • A. G. Ibrahim
Original Paper
  • 18 Downloads

Abstract

This work is interested in studying the motion of a rigid body carrying a rotor that rotates with a constant angular velocity about an axis parallel to the axis of dynamical symmetry. This motion is assumed to take place due to the effect of a combination of both uniform fields of gravity and magnetism that do not possess an axis of common symmetry. The equations of motion are constructed, and they are rewritten by means of the Hamiltonian function in the framework of the Lie–Poisson system. The equilibrium positions are inserted. The necessary conditions for the stability are introduced by applying the linear approximation method, while the sufficient conditions for stability are determined by utilizing the energy-Casimir method.

Notes

Acknowledgements

The authors acknowledge the Deanship of Scientific Research at King Faisal University for their support under Grant No. 17122010.

References

  1. 1.
    Rumiantsev, V.V.: Stability of permanent rotations of a heavy rigid body. Prikl. Math. Mekh. 20, 51–66 (1956)MathSciNetGoogle Scholar
  2. 2.
    Pozharitskii, G.K.: On the stability of permanent rotations of a rigid body with a fixed point under the action of a Newtonian central force field. J. Appl. Math. Mech. 23, 1134–1137 (1959)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Irtegov, V.D.: On the problem of stability of steady motions of a rigid body in a potential force field. J. Appl. Math. Mech. 30, 1113–1117 (1966)CrossRefzbMATHGoogle Scholar
  4. 4.
    Guliaev, M.P.: On the stability of rotations of a rigid body with one fixed point in the Euler case. Prikl. Math. Mech. 23, 579–582 (1959)Google Scholar
  5. 5.
    Lyapunov, A.M.: The General Problem of Stability of Motion. Obshch, Kharkov (1892)Google Scholar
  6. 6.
    Routh, E.J.: Dynamics of a System of Rigid Bodies: The Advanced Part. Dover Publications, New York (1955)zbMATHGoogle Scholar
  7. 7.
    Meirovitch, L.: Methods of Analytical Dynamics. McGraw-Hill, New York (1970)zbMATHGoogle Scholar
  8. 8.
    Rumiantsev, V.V.: On the stability of gyrostats. Prikl. Math. Mech. 25, 9–16 (1961)MathSciNetGoogle Scholar
  9. 9.
    Vera, J.A.: The gyrostat with a fixed point in a Newtonian force field: relative equilibria and stability. J. Math. Anal. Appl. 401, 836–849 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Guirao, J.L.G., Vera, J.A.: Equilibria, stability and Hamiltonian Hopf bifurcation of a gyrostat in an incompressible ideal fluid. Phys. D 241, 1648–1654 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Vera, A.J., Vigueras, A.: Hamiltonian dynamics of a gyrostat in the n-body problem: relative equilibria. Celest. Mech. Dyn. Astron. 94, 289–315 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Iñarrea, M., Lanchares, V., Pascual, A.I., Elipe, A.: Stability of the permanent rotations of an asymmetric gyrostat in a uniform Newtonian field. Appl. Math. Comput. 293, 404–415 (2017)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Elmandouh, A.A.: On the stability of the permanent rotations of a charged rigid body-gyrostat. Acta Mech. 228, 3947–3959 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Iñarrea, M., Lanchares, V., Pascual, A.I., Elipe, A.: On the stability of a class of permanent rotations of a heavy asymmetric gyrostat. Regul. Chaotic Dyn. 22, 824–839 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Tsogas, V., Kalvouridis, T.J., Mavraganis, A.G.: Equilibrium states of a gyrostat satellite in an annular configuration of N big bodies. Acta Mech. 175, 181–195 (2005)CrossRefzbMATHGoogle Scholar
  16. 16.
    Elipe, A., Lanchares, V.: Two equivalent problems: gyrostats in free motion and parametric quadratic Hamiltonians. Mech. Res. Commun. 24, 583–590 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kalvouridis, T.J., Tsogas, V.: Rigid body dynamics in the restricted ring problem of \(n+1\) bodies. Astrophys. Space Sci. 282, 749–763 (2002)CrossRefGoogle Scholar
  18. 18.
    Yehia, H.M., Elmandouh, A.A.: New conditional integrable cases of motion of a rigid body with Kovalevskaya’s configuration. J. Phys. A Math. Theor. 44, 012001 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    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, 142001 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Elmandouh, A.A.: New integrable problems in rigid body dynamics with quartic integrals. Acta Mech. 226, 2461–2472 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Elmandouh, A.A.: New integrable problems in the dynamics of particle and rigid body dynamics. Acta Mech. 226, 3749–3762 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Cantero, A., Crespo, F., Ferrer, S.: The triaxiality role in the spin-orbit dynamics of a rigid body. Appl. Math. Nonlinear Sci. 3, 187–208 (2018)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Crespo, F., Díaz-Toca, G., Ferrer, S., Lara, M.: Poisson and symplectic reductions of 4-DOF isotropic oscillators. The van der Waals system as benchmark. Appl. Math. Nonlinear Sci. 1, 473–492 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Doroshin, A.V.: Regimes of regular and chaotic motion of gyrostats in the central gravity field. Commun. Nonlinear Sci. Numer. Simul. 69, 416–431 (2019)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Chaikin, S.V.E.: The set of relative equilibria of a stationary orbital asymmetric gyrostat. Sib. Zhurnal Ind. Mat. 22, 116–121 (2019)Google Scholar
  26. 26.
    Chegini, M., Sadati, H., Salarieh, H.: Chaos analysis in attitude dynamics of a flexible satellite. Nonlinear Dyn. 93, 1421–1438 (2018)CrossRefzbMATHGoogle Scholar
  27. 27.
    Brun, F.: Rotation kring fix punkt. Ark. Mat. Ast. Fys. 6, 1–5 (1909)zbMATHGoogle Scholar
  28. 28.
    Bogoyavelensky, O.I.: New integrable problem of classical mechanics. Commun. Math. Phys. 94, 255–269 (1984)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Bogoyavelensky, O.I.: Euler equations on finite dimensional Lie algebra’s arising in physical problems. Commun. Math. Phys. 9, 307–315 (1984)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Yehia, H.M.: New integrable cases in the dynamics of rigid bodies. Mech. Res. Commun. 13, 169–172 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Yehia, H.M.: New integrable cases in the dynamics of rigid bodies, II. Mech. Res. Commun. 14, 1–56 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Bobenko, A.I., Reyman, A.G., Semenov-Tian-Shansky, M.A.: The Kowalewski top 99 years later: a lax pair, generalization and explicit solutions. Commun. Math. Phys. 122, 321–354 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Hassan, S.Z., Kharrat, B.N., Yehia, H.M.: On the stability of motion of a gyrostat about a fixed point under the action of non-symmetric fields. Eur. J. Mech. A/Solids 18, 313–318 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Volterra, V.: Sur la theories des variations des latitudes. Acta Math. 22, 201–357 (1899)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Gluhovsky, A., Christopher, T.: The structure of energy conserving low-order models. Phys. Fluid 11, 334–337 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Hughes, P.C.: Spacecraft Attitude Dynamics. Wiley, New York (1986)Google Scholar
  37. 37.
    Abouelmagd, E.I., Guirao, J.L., Hobiny, A., Alzahrani, F.: Dynamics of a tethered satellite with variable mass. Discrete Contin. Dyn. Syst. Ser. S 8, 1035–1045 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Abouelmagd, E.I., Guirao, J.L., Hobiny, A., Alzahrani, F.: Stability of equilibria points for a dumbbell satellite when the central body is oblate spheroid. Discrete Contin. Dyn. Syst. Ser. S 8, 1047–1054 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Abouelmagd, E.I., Guirao, J.L., Vera, J.A.: Dynamics of a dumbbell satellite under the zonal harmonic effect of an oblate body. Commun. Nonlinear Sci. Numer. Simul. 20, 1057–1069 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Leimanis, E.: The General Problem of Motion of Coupled Rigid Bodies About a Fixed Point. Springer, Berlin (1965)CrossRefzbMATHGoogle Scholar
  41. 41.
    Yehia, H.M.: Equivalent problems in rigid body dynamics—I. Celest. Mech. 41, 275–288 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Leonard, N.E.: Stability of a bottom-heavy underwater vehicle. Automatica 33, 331–346 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

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

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