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On the orbital stability of the motion of a rigid body in the case of Bobylev–Steklov

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Abstract

The problem of motion of the heavy rigid body about a fixed point admits simple periodic solutions in few cases. Examples are the pendulum-like plane motions, Grioli’s case and Bobylev–Steklov case. Noting that only stable motions can be realized due to the inevitable deviations in the initial conditions and in the determination of the distribution of mass in the body, the study of stability acquires an increasing importance. The stability of plane motions was considered in several works. Grioli’s case was studied recently by Markeyev. The aim of the present work is to study stability in the linear approximation for Bobylev and Steklov’s case. The use of Euler–Poisson equations and their integrals for the study of stability of periodic motions is quite complicated. Instead, we use a single second-order differential equation obtained by one of us, by the maximal reduction of the order of equations of motion using their general integrals. This equation is satisfied by the trajectory of the trace of the vertical on the Poisson sphere fixed in the body. The orbital stability of a solution means that the perturbed trajectory remains near to the unperturbed, after perturbations preserving the values of general integrals. After classification of the two possible families of trajectories, equation in variation is obtained for each family. In the three-dimensional space of parameters affecting stability, we determine the surfaces carrying primitive periodic solutions, and thus separating stability and instability zones. Both equations in the variations were solved also numerically on certain sections of the parameter space. Numerical results accomplish the identification of zones lying between surfaces as stability or instability zones and do not show any traces of other zones, rather than those detected by analytical study.

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Acknowledgments

The authors thank two anonymous referees for their remarks, which have helped better presentation of the paper.

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Authors and Affiliations

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

    H. M. Yehia, S. Z. Hassan & M. E. Shaheen

  2. Department of Mathematics, Faculty of Education, Dammam University, Al-Jubail, 12020, Saudi Arabia

    S. Z. Hassan

  3. Department of Mathematics, Port Said University, Port Said, Egypt

    M. E. Shaheen

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  1. H. M. Yehia
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  3. M. E. Shaheen
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Correspondence to H. M. Yehia.

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Yehia, H.M., Hassan, S.Z. & Shaheen, M.E. On the orbital stability of the motion of a rigid body in the case of Bobylev–Steklov. Nonlinear Dyn 80, 1173–1185 (2015). https://doi.org/10.1007/s11071-015-1934-3

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