Abstract
We deal with the stability problem of resonant rotation of a symmetric rigid body about its center of mass in an elliptical orbit. The resonant rotation is a planar motion such that the body completes one rotation in absolute space during two orbital revolutions of its center of mass. In [1–3] the stability analysis of the above resonant rotation with respect to planar perturbations has been performed in detail.
In this paper we study the stability of the resonant rotation in an extended formulation taking into account both planar and spatial perturbations. By analyzing linearized equations of perturbed motion, we found eccentricity intervals, where the resonant rotation is unstable. Outside of these intervals a nonlinear stability study has been performed and subintervals of formal stability and stability for most initial data have been found. In addition, the instability of the resonant rotation was established at several eccentricity values corresponding to the third and fourth order resonances.
Our study has also shown that in linear approximation the spatial perturbations have no effect on the stability of the resonant rotation, whereas in a nonlinear system they can lead to its instability at some resonant values of the eccentricity.
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References
Khentov, A. A., On Rotational Motion of a Satellite, Kosmicheskie Issledovaniya, 1984, vol. 22, no. 1, pp. 130–131 (Russian).
Markeev, A.P. and Bardin, B. S., A Planar, Rotational Motion of a Satellite in an Elliptic Orbit, Cosmic Research, 1994, vol. 32, no. 6, pp. 583–589; see also: Kosmicheskie Issledovaniya, 1994, vol. 32, no. 6, pp. 43–49.
Bardin, B. S., Chekina, E. A., and Chekin, A.M., On the Stability of a Planar Resonant Rotation of a Satellite in an Elliptic Orbit, Regul. Chaotic Dyn., 2015, vol. 20, no. 1, pp. 63–73.
Beletskii, V. V. and Shlyakhtin, A. N., Resonance Rotations of a Satellite with Interactions between Magnetic and Gravitational Fields, Preprint No. 46, Moscow: Institute of Applied Mathematics, Academy of Sciences of the USSR, 1980 (Russian).
Malkin, I.G., Theory of Stability of Motion, Moscow: Nauka, 1952 (Russian); see also: Ann Arbor,Mich.: Univ. of Michigan Library, 1958.
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).
Markeev, A.P., Stability of Equilibrium States of Hamiltonian Systems: A Method of Investigation, Mech. Solids, 2004, vol. 39, no. 6, pp. 1–8; see also: Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, 2004, vol. 39, no. 6, pp. 3–12.
Markeyev, A.P., A Constructive Algorithm for the Normalization of a Periodic Hamiltonian, J. Appl. Math. Mech., 2005, vol. 69, no. 3, pp. 323–337; see also: Prikl. Mat. Mekh., 2005, vol. 69, no. 3, pp. 355–371.
Markeev, A.P., Libration Points in Celestial Mechanics and Space Dynamics, Moscow: Nauka, 1978 (Russian).
Arnol’d, V. I., Mathematical Methods of Classical Mechanics, 2nd ed., Grad. Texts in Math., vol. 60, New York: Springer, 1989.
Glimm, J., Formal Stability of Hamiltonian Systems, Comm. Pure Appl. Math., 1964, vol. 17, no. 4, pp. 509–526.
Moser, J., New Aspects in the Theory of Stability of Hamiltonian Systems, Comm. Pure Appl. Math., 1958, vol. 11, no. 1, pp. 81–114.
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Bardin, B.S., Chekina, E.A. On the stability of resonant rotation of a symmetric satellite in an elliptical orbit. Regul. Chaot. Dyn. 21, 377–389 (2016). https://doi.org/10.1134/S1560354716040018
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DOI: https://doi.org/10.1134/S1560354716040018