Skip to main content
SpringerLink
Log in

On the Orbital Stability of Pendulum Motions of a Rigid Body in the Hess Case

  • MATHEMATICS
  • Published:
Doklady Mathematics Aims and scope Submit manuscript

Abstract

Given a heavy rigid body with one fixed point, we investigate the problem of orbital stability of its periodic motions. Based on the analysis of the linearized system of equations of perturbed motion, the orbital instability of the pendulum rotations is proved. In the case of pendulum oscillations, a transcendental situation occurs, when the question of stability cannot be solved using terms of an arbitrarily high order in the expansion of the Hamiltonian of the equations of perturbed motion. It is proved that the pendulum oscillations are orbitally unstable for most values of the parameters.

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

Similar content being viewed by others

REFERENCES

  1. W. Hess, “Über die Eulerschen bewegungsgleichungen und über eine neue partikuläre Lösung des Problems der Bewegung eines starren Körpers um einen festen Punkt,” Math. Ann. 37 (2), 153–181 (1890).

    Article  MathSciNet  Google Scholar 

  2. A. P. Markeev, “The stability of the plane motions of a rigid body in the Kovalevskaya case,” J. Appl. Math. Mech. 65 (1), 47–54 (2001).

    Article  MathSciNet  Google Scholar 

  3. A. P. Markeev, S. V. Medvedev, and T. N. Chekhovskaya, “The problem of stability of pendulum-like vibrations of a rigid body in Kovalevskaya’s case,” Mech. Solids 38 (1), 1–6 (2003).

    Google Scholar 

  4. A. P. Markeev, “The pendulum-like motions of a rigid body in the Goryachev–Chaplygin case,” J. Appl. Math. Mech. 68 (2), 249–258 (2004).

    Article  MathSciNet  Google Scholar 

  5. B. S. Bardin, “On the orbital stability of pendulum-like motions of a rigid body in the Bobylev–Steklov case,” Regular Chaotic Dyn. 15 (6), 702–714 (2010).

    Article  MathSciNet  Google Scholar 

  6. B. S. Bardin, T. V. Rudenko, and A. A. Savin, “On the orbital stability of planar periodic motions of a rigid body in the Bobylev–Steklov case,” Regular Chaotic Dyn. 17 (6), 533–546 (2012).

    Article  MathSciNet  Google Scholar 

  7. B. S. Bardin and A. A. Savin, “On the orbital stability of pendulum-like oscillations and rotations of a symmetric rigid body with a fixed point,” Regular Chaotic Dyn. 17 (3–4), 243–257 (2012).

    Article  MathSciNet  Google Scholar 

  8. B. S. Bardin and A. A. Savin, “The stability of the plane periodic motions of a symmetrical rigid body with a fixed point,” J. Appl. Math. Mech. 77 (6), 578–587 (2013).

    Article  MathSciNet  Google Scholar 

  9. B. S. Bardin and E. A. Chekina, “On the orbital stability of pendulum-like oscillations of a heavy rigid body with a fixed point in the Bobylev–Steklov case,” Russ. J. Nonlinear Dyn. 17 (4), 453–464 (2021).

    MathSciNet  Google Scholar 

  10. G. E. O. Giacaglia, Perturbation Methods in Non-Linear System (Springer, New York, 1972).

    Book  Google Scholar 

  11. A. P. Markeev, Libration Points in Celestial Mechanics and Astrodynamics (Fizmatlit, Moscow, 1978) [in Russian].

    Google Scholar 

  12. C. L. Siegel and J. K. Moser, Lectures on Hamiltonian Systems (Springer, Heidelberg, 1971).

    Google Scholar 

  13. V. I. Arnold, “Small denominators and problems of stability of motion in classical and celestial mechanics,” Russ. Math. Surv. 18 (6), 85–191 (1963).

    Article  MathSciNet  Google Scholar 

  14. V. D. Irtegov, “Stability of pendulum-like oscillations of the Kovalevskaya gyroscope,” Tr. Kazan. Aviats. Inst. Mat. Mekh. 97, 38–40 (1968).

    Google Scholar 

  15. A. Z. Bryum, “Orbital stability analysis using first integrals,” J. Appl. Math. Mech. 53 (6), 689–695 (1989).

    Article  MathSciNet  Google Scholar 

  16. A. V. Bolsinov, A. V. Borisov, and I. S. Mamaev, “Topology and stability of integrable systems,” Russ. Math. Surv. 65 (2), 259–318 (2010).

    Article  MathSciNet  Google Scholar 

  17. G. K. Suslov, Theoretical Mechanics (Gostekhizdat, Moscow, 1946) [in Russian].

    Google Scholar 

  18. A. M. Lyapunov, “General problem of the stability of motion,” Collected Works (Akad. Nauk SSSR, Moscow, 1956), Vol. 2, pp. 7–263 [in Russian].

    Google Scholar 

  19. I. N. Gashenenko, “Poinsot kinematic representation of the motion of a body in the Hess case,” Mekh. Tverd. Tela 40, 12–20 (2010).

    MathSciNet  Google Scholar 

  20. B. S. Bardin, “On the stability of a periodic Hamiltonian system with one degree of freedom in a transcendental case,” Dokl. Math. 97 (2), 161–163 (2018).

    Article  MathSciNet  Google Scholar 

Download references

Funding

This work was carried out in Moscow Aviation Institute (National Research University) and was supported by the Russian Science Foundation, project no. 22-21-00729.

Author information

Authors and Affiliations

  1. Moscow Aviation Institute (National Research University), Moscow, Russia

    B. S. Bardin & A. A. Savin

Authors
  1. B. S. Bardin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. A. Savin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to B. S. Bardin or A. A. Savin.

Ethics declarations

The authors of this work declare that they have no conflicts of interest.

Additional information

Translated by I. Ruzanova

Publisher’s Note.

Pleiades Publishing 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

Bardin, B.S., Savin, A.A. On the Orbital Stability of Pendulum Motions of a Rigid Body in the Hess Case. Dokl. Math. 109, 52–55 (2024). https://doi.org/10.1134/S1064562424701795

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1064562424701795

Keywords:

Search

Navigation