Abstract
We deal with the problem of orbital stability of pendulum-like periodic motions of a heavy rigid body with a fixed point. We suppose that a mass geometry corresponds to the Bobylev-Steklov case. The stability problem is solved in nonlinear setting.
In the case of small amplitude oscillations and rotations with large angular velocities the small parameter can be introduced and the problem can be investigated analytically.
In the case of unspecified oscillation amplitude or rotational angular velocity the problem is reduced to analysis of stability of a fixed point of the symplectic map generated by the equations of the perturbed motion. The coefficients of the symplectic map are determined numerically. Rigorous results on the orbital stability or instability of unperturbed motion are obtained by analyzing these coefficients.
Similar content being viewed by others
References
Irtegov, V. D., The Stability of the Pendulum-Like Oscillations of a Kovalevskaya Gyroscope, Tr. Kazan. Aviats. Inst., 1968, vol. 97, pp. 38–40 (Russian).
Bryum, A. Z., A Study of Orbital Stability by Means of First Integrals, Prikl. Mat. Mekh., 1989, vol. 53, no. 6, pp. 873–879 [J. Appl. Math. Mech., 1989, vol. 53, no. 6, pp. 689–695].
Markeev, A.P., On the Stability of the Planar Motions of a Rigid Body in the Kovalevskaya Case, Prikl. Mat. Mekh., 2001, vol. 65, no. 1, pp. 51–58 [J. Appl. Math. Mech., 2001, vol. 65, no. 1, pp. 47–54].
Markeev, A.P., Medvedev, S.V., and Chekhovskaya, T. N., To the Problem of Stability of Pendulum-like Vibrations of a Rigid Body in Kovalevskaya’s Case, Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, 2003, vol. 1, pp. 3–9 [Mech. Solids, 2003, vol. 38, no. 1, pp. 1–6].
Markeev, A.P., On the Identity Resonance in a Particular Case of the problem of Stability of Periodic Motions of a Rigid Body, Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, 2003, vol. 3, pp. 32–37 [Mech. Solids, 2003, vol. 38, no. 3, pp. 22–26].
Markeev, A.P., The Pendulum-Like Motions of a Rigid Body in the Goryachev-Chaplygin Case, Prikl. Mat. Mekh., 2004, vol. 68, no. 2, pp. 282–293 [J. Appl. Math. Mech., 2004, vol. 68, no. 2, pp. 249–258].
Alekhin, A.K., On the Stability of Plane Motions of a Heavy Axially Symmetric Rigid Body, Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, 2006, vol. 4, pp. 56–62 [Mech. Solids, 2006, vol. 41, no. 4, pp. 40–45].
Bardin, B. S., Stability Problem for Pendulum-Type Motions of a Rigid Body in the Goryachev-Chaplygin Case, Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, 2007, vol. 2, pp. 14–21 [Mech. Solids, 2007, vol. 42, vol. 2, pp. 177–183].
Bardin, B. S., On Orbital Stability of Pendulum-Like Motions of a Rigid Body in the Bobylev-Steklov Case, Regul. Chaotic Dyn., 2010, vol. 15. no. 6, pp. 704–716.
Bardin, B. S. and Savin, A. A., On Orbital Stability Pendulum-Like Oscillations and Rotation of Symmetric Rigid Body with a Fixed Point, Regul. Chaotic Dyn., 2012, vol. 17. nos. 3–4, pp. 243–257.
Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Topology and Stability of Integrable Systems, Uspekhi Mat. Nauk, 2010, vol. 65, no. 2, pp. 71–132 [Russian Math. Surveys, 2010, vol. 65, no. 2, pp. 259–318].
Bobylev, D.N., On a Particular Solution of the Differential Equations of a Heavy Rigid Body Rotation Around of a Fixed Point, Tr. Fiz. Obshch. Lyubitelei Yestestv., 1896, vol. 8, no. 2, pp. 21–25 (Russian).
Steklov, V. A., A Case of Motion of a Rigid Body with a Fixed Point, Tr. Fiz. Obshch. Lyubitelei Yestestv., 1896, vol. 8, no. 2, pp. 19–21 (Russian).
Kuzmin, P.A., Particular Cases of Motion of a Heavy Rigid Body About a Fixed Point (in Works of Russian Authors), Tr. Kazan. Aviats. Inst., 1953, vol. 27, pp. 91–121 (Russian).
Dokshevich, A. I., Solutions in a Finite Form of the Euler-Poisson Equations, Kiev: Naukova Dumka, 1992 (Russian).
Borisov, A.V. and Mamaev, I. S., Rigid Body Dynamics: Hamiltonian Methods, Integrability, Chaos, Moscow-Izhevsk: Institute of Computer Science, 2005 (Russian).
Akhiezer, N. I., Elements of the Theory of Elliptic Functions, 2nd ed., Moscow: Nauka, 1970 [Providence, RI: AMS, 1990].
Markeev, A.P., Libration Points in Celestial Mechanics and Space Dynamics, Moscow: Nauka, 1978 (Russian).
Malkin, I.G., Theory of Stability of Motion, 2nd ed., Moscow: Nauka, 1966 (Russian).
Gradshteyn, I. S. and Ryzhik, I.M., Table of Integrals, Series, and Products, Moscow: Fizmatlit, 1963 [6th ed., San Diego, CA: Acad. Press, 2000].
Yakubovich, V.A. and Starzhinskii, V.M., Parametric Resonance in Linear Systems, Moscow: Nauka, 1987 (Russian).
Markeev, A. P., Linear Hamiltonian Systems and Some Problems of Stability of the Satellite Center of Mass, Moscow-Izhevsk: R&C Dynamics, Institute of Computer Science, 2009 (Russian).
Birkhoff, G.D., Dynamical Systems, Amer. Math. Soc. Colloq. Publ., vol. 9, Providence, R. I.: AMS, 1966.
Giacaglia, G.E.O., Perturbation Methods in Non-Linear Systems, New York: Springer, 1972.
Ivanov, A.P. and Sokol’skii, A. G., On the Stability of a Nonautonomous Hamiltonian System under a Parametric Resonance of Essential Type, Prikl. Mat. Mekh., 1980, vol. 44, no. 6, pp. 963–970 [J. Appl. Math. Mech., 1980, vol. 44, no. 6, pp. 687–691].
Arnol’d, V. I., Small Denominators and Problems of Stability of Motion in Classical and Celestial Mechanics, Uspekhi Mat. Nauk, 1963, vol. 18, no. 6(114), pp. 91–192 [Russian Math. Surveys, 1963, vol. 18, no. 6, pp. 85–191].
Moser, J., Lectures on Hamiltonian Systems, Mem. Amer. Math. Soc., vol. 81, Providence, RI: AMS, 1968.
Markeev, A.P., Stability of Equilibrium States of Hamiltonian Systems: A Method of Investigation, Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, 2004, vol. 39, no. 6, pp. 3–12 [Mech. Solids, 2004, vol. 39, no. 6, pp. 1–8].
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bardin, B.S., Rudenko, T.V. & Savin, A.A. On the orbital stability of planar periodic motions of a rigid body in the Bobylev-Steklov case. Regul. Chaot. Dyn. 17, 533–546 (2012). https://doi.org/10.1134/S1560354712060056
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1560354712060056