On Nonlinear Resonant Oscillations of a Rigid Body Generated by Its Conical Precession

    Received 26 July 2018; accepted 24 August 2018

    2018, Vol. 14, no. 4, pp.  503-518

    Author(s): Markeev A. P.

    The motion of a dynamically symmetric rigid body relative to its center of mass in the central Newtonian gravitational field in a circular orbit is investigated. This problem involves motion (called conical precession) where the dynamical symmetry axis of the body is located all the time in the plane perpendicular to the velocity vector of the center of mass of the body and makes a constant angle with the direction of the radius vector of the center of mass relative to the attracting center. This paper deals with a special case in which this angle is $\pi/4$ and the ratio between the polar and the equatorial principal central moments of inertia of the body is equal to the number $2/3$ or is close to it. In this case, the conical precession is stable with respect to the angles that define the position of the symmetry axis in an orbital coordinate system and with respect to the time derivatives of these angles, and the frequencies of small (linear) oscillations of the symmetry axis are equal or close to each other (that is, the 1:1 resonance takes place). Using classical perturbation theory and modern numerical and analytical methods of nonlinear dynamics, a solution is presented to the problem of the existence, bifurcations and stability of periodic motions of the symmetry axis of a body which are generated from its relative (in the orbital coordinate system) equilibrium corresponding to conical precession. The problem of the existence of conditionally periodic motions is also considered.
    Keywords: resonance, stability, oscillations, canonical transformations
    Citation: Markeev A. P., On Nonlinear Resonant Oscillations of a Rigid Body Generated by Its Conical Precession, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 4, pp.  503-518
    DOI:10.20537/nd180406


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    References
    [1] Beletskii, V. V., Motion of an Artificial Satellite about Its Center of Mass, Jerusalem, 1966, x, 261 pp.
    [2] Beletskii, V. V., Satellite's Motion about Center of Mass in a Gravitational Field, MGU, Moscow, 1975 (Russian)  adsnasa
    [3] Markeev, A. P., Linear Hamiltonian Systems and Some Problems of Stability of the Satellite Center of Mass, R&C Dynamics, Institute of Computer Science, Izhevsk, 2009, 396 pp. (Russian)
    [4] Korteweg, D. J., “Sur certaines vibrations d'ordre supérieur et d'intensité anormale — vibrations de relations, — dans les mécanismes à plusieurs degrés de liberté”, Arch. Néerl. sci. exactes et natur. Sér. 2, 1 (1898), 229–260  zmath
    [5] Beth, H. I. E., “Les oscillations autour d'une position dans le cas d'existence d'une relation linéaire simple entre les nombres vibratoires”, Arch. Néerl. sci. exactes et natur. Sér. 2, 15 (1910), 246–283  zmath
    [6] Beth, H. I. E., “Les oscillations autour d'une position dans le cas d'existence d'une relation linéaire simple entre les nombres vibratoires (suite)”, Arch. Néerl. sci. exactes et natur. Sér. 3A, 1 (1912), 185–213  adsnasa
    [7] Markeyev, A. P., “Non-Linear Oscillations of a $1:1$ Resonance Hamiltonian System”, J. Appl. Math. Mech., 75:6 (2011), 631–646  crossref  mathscinet; Prikl. Mat. Mekh., 75:6 (2011), 901–922 (Russian)  mathscinet
    [8] Hénon, M. and Heiles, C., “The Applicability of the Third Integral of Motion: Some Numerical Experiments”, Astronom. J., 69:1 (1964), 73–79  mathscinet  adsnasa
    [9] Roels, J., “An Extension to Resonant Cases of Liapunov's Theorem Concerning the Periodic Solutions near a Hamiltonian Equilibrium”, J. Differential Equations, 9:2 (1971), 300–324  crossref  mathscinet  zmath  adsnasa
    [10] Braun, M., “On the Applicability of the Third Integral of Motion”, J. Differential Equations, 13:2 (1973), 300–318  crossref  mathscinet  zmath  adsnasa
    [11] Breiter, S. and Elipe, A., “Pseudo-Oscillator with a Quartic Perturbation”, Mech. Res. Comm., 28:2 (2001), 119–126  crossref  mathscinet  zmath
    [12] Kholostova, O. V. and Safonov, A. I., “Investigation of the Motions of an Autonomous Hamiltonian System at a $1:1$ Resonance”, Regul. Chaotic Dyn., 22:7 (2017), 792–807  mathnet  crossref  mathscinet  zmath  adsnasa
    [13] Birkhoff, G. D., Dynamical Systems, AMS, Providence, R.I., 1966, 305 pp.  mathscinet  zmath
    [14] Giacaglia, G. E. O., Perturbation Methods in Non-Linear Systems, Appl. Math. Sci., 8, Springer, New York, 1972, viii+369 pp.  crossref  mathscinet  zmath
    [15] Markeyev, A. P., “The Critical Case of a Pair of Zero Roots in a Two-Degree-of-Freedom Hamiltonian System”, J. Appl. Math. Mech., 62:3 (1998), 341–349  crossref  mathscinet; Prikl. Mat. Mekh., 62:3 (1998), 372–382 (Russian)  mathscinet
    [16] Markeev, A. P., “On Stability and Non-Linear Oscillations of Hamiltonian System in a Resonant Case”, Mech. Solids, 33:4 (1998), 32–41; Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, 1998, no. 4, 38–49 (Russian)
    [17] Arnol'd, V. I., Kozlov, V. V., and Neĭshtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, Encyclopaedia Math. Sci., 3, 3rd ed., Springer, Berlin, 2006, xiv+518 pp.  mathscinet  zmath
    [18] Markeev, A. P., “Nonlinear Oscillations of Sympathetic Pendulums”, Nelin. Dinam., 6:3 (2010), 605–621 (Russian)  mathnet  crossref
    [19] Markeyev, A. P., “An Algorithm for Normalizing Hamiltonian Systems in the Problem of the Orbital Stability of Periodic Motions”, J. Appl. Math. Mech., 66:6 (2002), 889–896  crossref  mathscinet; Prikl. Mat. Mekh., 66:6 (2002), 929–938 (Russian)  mathscinet
    [20] Neishtadt, A. I., “Estimates in the Kolmogorov Theorem on Conservation of Conditionally Periodic Motions”, J. Appl. Math. Mech., 45:6 (1981), 766–772  crossref  mathscinet; Prikl. Mat. Mekh., 45:6 (1981), 1016–1025 (Russian)  mathscinet



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