Skip to main content
SpringerLink
Log in

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

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Rumiantsev, V.V.: Stability of permanent rotations of a heavy rigid body. Prikl. Math. Mekh. 20, 51–66 (1956)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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. Lyapunov, A.M.: The General Problem of Stability of Motion. Obshch, Kharkov (1892)

    Google Scholar 

  6. Routh, E.J.: Dynamics of a System of Rigid Bodies: The Advanced Part. Dover Publications, New York (1955)

    MATH  Google Scholar 

  7. Meirovitch, L.: Methods of Analytical Dynamics. McGraw-Hill, New York (1970)

    MATH  Google Scholar 

  8. Rumiantsev, V.V.: On the stability of gyrostats. Prikl. Math. Mech. 25, 9–16 (1961)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  13. Elmandouh, A.A.: On the stability of the permanent rotations of a charged rigid body-gyrostat. Acta Mech. 228, 3947–3959 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  16. Elipe, A., Lanchares, V.: Two equivalent problems: gyrostats in free motion and parametric quadratic Hamiltonians. Mech. Res. Commun. 24, 583–590 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  20. Elmandouh, A.A.: New integrable problems in rigid body dynamics with quartic integrals. Acta Mech. 226, 2461–2472 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Elmandouh, A.A.: New integrable problems in the dynamics of particle and rigid body dynamics. Acta Mech. 226, 3749–3762 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  25. Chaikin, S.V.E.: The set of relative equilibria of a stationary orbital asymmetric gyrostat. Sib. Zhurnal Ind. Mat. 22, 116–121 (2019)

    MathSciNet  Google Scholar 

  26. Chegini, M., Sadati, H., Salarieh, H.: Chaos analysis in attitude dynamics of a flexible satellite. Nonlinear Dyn. 93, 1421–1438 (2018)

    Article  MATH  Google Scholar 

  27. Brun, F.: Rotation kring fix punkt. Ark. Mat. Ast. Fys. 6, 1–5 (1909)

    MATH  Google Scholar 

  28. Bogoyavelensky, O.I.: New integrable problem of classical mechanics. Commun. Math. Phys. 94, 255–269 (1984)

    Article  MathSciNet  Google Scholar 

  29. Bogoyavelensky, O.I.: Euler equations on finite dimensional Lie algebra’s arising in physical problems. Commun. Math. Phys. 9, 307–315 (1984)

    Article  MathSciNet  Google Scholar 

  30. Yehia, H.M.: New integrable cases in the dynamics of rigid bodies. Mech. Res. Commun. 13, 169–172 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  31. Yehia, H.M.: New integrable cases in the dynamics of rigid bodies, II. Mech. Res. Commun. 14, 1–56 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  34. Volterra, V.: Sur la theories des variations des latitudes. Acta Math. 22, 201–357 (1899)

    Article  MathSciNet  MATH  Google Scholar 

  35. Gluhovsky, A., Christopher, T.: The structure of energy conserving low-order models. Phys. Fluid 11, 334–337 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  36. Hughes, P.C.: Spacecraft Attitude Dynamics. Wiley, New York (1986)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  40. Leimanis, E.: The General Problem of Motion of Coupled Rigid Bodies About a Fixed Point. Springer, Berlin (1965)

    Book  MATH  Google Scholar 

  41. Yehia, H.M.: Equivalent problems in rigid body dynamics—I. Celest. Mech. 41, 275–288 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  42. Leonard, N.E.: Stability of a bottom-heavy underwater vehicle. Automatica 33, 331–346 (1997)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Faculty of Science, King Faisal University, P. O. Box 400, Al-Ahsa, 31982, Saudi Arabia

    A. A. Elmandouh & A. G. Ibrahim

  2. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt

    A. A. Elmandouh

Authors
  1. A. A. Elmandouh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. G. Ibrahim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Elmandouh.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Elmandouh, A.A., Ibrahim, A.G. Hamiltonian structure, equilibria, and stability for an axisymmetric gyrostat motion in the presence of gravity and magnetic fields. Acta Mech 230, 2539–2548 (2019). https://doi.org/10.1007/s00707-019-02413-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-019-02413-y

Search

Navigation