Research on the Motion of a Body in a Potential Force Field in the Case of Three Invariant Relations

    Received 25 April 2019

    2019, Vol. 15, no. 3, pp.  327-342

    Author(s): Gorr G. V., Tkachenko D., Shchetinina E. K.

    The problem of the motion of a rigid body with a fixed point in a potential force field is considered. A new case of three nonlinear invariant relations of the equations of motion is presented. The properties of Euler angles, Rodrigues – Hamilton parameters, and angular velocity hodographs in the Poinsot method are investigated using an integrated approach in the interpretation of body motion.
    Keywords: potential force field, Euler angles, Rodrigues – Hamilton parameters, Poinsot method
    Citation: Gorr G. V., Tkachenko D., Shchetinina E. K., Research on the Motion of a Body in a Potential Force Field in the Case of Three Invariant Relations, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 3, pp.  327-342
    DOI:10.20537/nd190310


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    References

    [1] Zhukovskiy, N. E., “On Importance of the Geometrical Interpretation in the Theoretical Mechanics”, Collected Works, v. 7, Gostekhizdat, Moscow, 1950, 9–15 (Russian)  mathscinet
    [2] Kharlamov, P. V., “New Methods in the Dynamics of Systems of Rigid Bodies”, Dynamics of Multibody Systems: International Union of Theoretical and Applied Mechanics Proc., ed. K. Magnus, Springer, Berlin, 1978, 133–143  crossref  mathscinet
    [3] Leimanis, E., The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point, Springer Tracts in Natural Philosophy, 7, Springer, Berlin, 1965, 356 pp.  crossref
    [4] Gorr, G. V., Kudryashova, L. V., and Stepanova, L. A., Classical Problems in the Theory of Solid Bodies, Their Development and Current State, Naukova Dumka, Kiev, 1978, 294 pp. (Russian)  mathscinet
    [5] Borisov, A. V. and Mamaev, I. S., Rigid Body Dynamics: Hamiltonian Methods, Integrability, Chaos, R&C Dynamics, Institute of Computer Science, Izhevsk, 2005, 576 pp. (Russian)  mathscinet
    [6] Gashenenko, I. N., Gorr, G. V., and Kovalev, A. M., Classical Problems of Rigid Body Dynamics, Naukova Dumka, Kiev, 2012, 402 pp. (Russian)  mathscinet  zmath
    [7] Kharlamov, P. V., “Current State and Develoment Prospects of Classical Problems of Rigid Body Dynamics”, Mekh. Tverd. Tela, 30 (2000), 1–12 (Russian)  mathscinet
    [8] Kovalev, A. M. and Gorr, G. V., “Donetsk School of Mechanics and Stability”, PNAES, 20:2(42) (2014), 96–126
    [9] Ziglin, S. L., “Branching of Solutions and Nonexistence of First Integrals in Hamiltonian Mechanics: 1”, Funct. Anal. Appl., 16:3 (1982), 181–189  mathnet  crossref  mathscinet; Funktsional. Anal. i Prilozhen., 16:3 (1982), 30–41 (Russian)  mathscinet; Ziglin, S. L., “Branching of Solutions and Nonexistence of First Integrals in Hamiltonian Mechanics: 2”, Funct. Anal. Appl., 17:1 (1983), 6–17  mathnet  crossref  mathscinet  zmath; Funktsional. Anal. i Prilozhen., 17:1 (1983), 8–23 (Russian)  crossref  mathscinet  zmath
    [10] Ziglin, S. L., “Splitting of the Separatrices and the Nonexistence of First Integrals in Systems of Differential Equations of Hamiltonian Type with Two Degrees of Freedom”, Math. USSR-Izv., 31:2 (1988), 407–421  mathnet  crossref  mathscinet  zmath; Izv. Akad. Nauk SSSR Ser. Mat., 51:5 (1987), 1088–1103, 1118–1119 (Russian)
    [11] Kozlov V. V., Onishchenko D. A., “Nonintegrability of Kirchhoff's Equations”, Sov. Math. Dokl., 26 (1982), 495–498  mathscinet  zmath; Dokl. Akad. Nauk SSSR, 266:6 (1982), 1298–1300 (Russian)  mathnet  mathscinet  zmath
    [12] Borisov, A. V., “Necessary and Sufficient Conditions for Integrability of Kirchhoff Equations”, Regul. Chaotic Dyn., 1:2 (1996), 61–76 (Russian)  mathnet  mathscinet  zmath
    [13] Burov, A. A., “Nonintegrability of the Equations of Gyrostat Motions in Cardan Suspension”, Research Problems of Stability and Stabilisation of Motion, Computing Centre of the USSR Acad. Sci., Moscow, 1986, 3–10 (Russian)  mathscinet  adsnasa
    [14] Bogoyavlenskii, O. I., “Integrable Problems of the Dynamics of Coupled Rigid Bodies”, Russian Acad. Sci. Izv. Math., 41:3 (1993), 395–416  mathnet  mathscinet; Izv. Ross. Akad. Nauk Ser. Mat., 56:6 (1992), 1139–1164 (Russian)
    [15] Gorr, G. V., Invariant Relations of Equations of Rigid Body Dynamics: Theory, Results, Comments, R&C Dynamics, Institute of Computer Science, Izhevsk, 2017, 424 pp. (Russian)  mathscinet
    [16] Poincaré, H., Les méthodes nouvelles de la mécanique céleste: In 3 Vols., Gauthier-Villars, Paris; Dover, New York, 1892, 1893, 1899  mathscinet  adsnasa
    [17] Levi-Civita, T. and Amaldi, U., Lezioni di meccanica razionale: Vol. 1. Cinematica — principi e statica, Zanichelli, Bologna, 1950, xviii+816 pp.  mathscinet; Levi-Civita, T. and Amaldi, U., Lezioni di meccanica razionale: Vol. 2. Dinamica dei sistemi con un numero finito di gradi di liberta, Zanichelli, Bologna, 1951, 1952  mathscinet
    [18] Chaplygin, S. A., “On the Last Multiplier Principle”, Mat. Sb., 21:3 (1900), 479–489 (Russian)  mathnet
    [19] Kharlamov, P. V., “On Invariant Relations of a System of Ordinary Differential Equations”, Mekh. Tverd. Tela, 6 (1974), 15–24 (Russian)  zmath
    [20] Poinsot, L., “Théorie nouvelle de la rotation des corps”, J. Math. Pures Appl., 16 (1851), 9–-129, 289–336
    [21] Routh, E. J., The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies: Being Part II of a Treatise on the Whole Subject, 6th ed., Dover, New York, 1955, xiv+484 pp.  mathscinet
    [22] Gorr, G. V., “On an Approach in the Use of the Poinsot Theorem of kinematic Interpretation of the Body Motion with Fixed Point”, Mekh. Tverd. Tela, 42 (2012), 26–36 (Russian)  mathscinet  zmath
    [23] Yehia, H. M., “Transformations of Mechanical Systems with Cyclic Coordinates and New Integrable Problems”, J. Phys. A, 34:49 (2001), 11167–11183  crossref  mathscinet  zmath  adsnasa
    [24] Yehia, H. M., “New Solvable Problems in the Dynamics of a Rigid Body about a Fixed Point in a Potential Field”, Mech. Res. Commun., 57 (2014), 44–48  crossref  mathscinet  elib
    [25] Ol'shanskiy, V. Yu., “Linear Invariant Relations of Kirchhoff's Equations”, J. Appl. Math. Mech., 79:4 (2015), 334–349  crossref  mathscinet; Prikl. Mat. Mekh., 79:4 (2015), 476–497 (Russian)
    [26] Ol'shanskiy, V. Yu., “A New Linear Invariant Relation of the Poincaré – Zhukovskii Equations”, J. Appl. Math. Mech., 76:6 (2012), 636–645  crossref  mathscinet; Prikl. Mat. Mekh., 76:6 (2012), 883–894 (Russian)  mathscinet
    [27] Mukharlyamov, R. G., “Stabilization of the Motions of Mechanical Systems in Prescribed Phase-Space Manifolds”, J. Appl. Math. Mech., 70:2 (2006), 210–222  crossref  mathscinet  zmath  elib; Prikl. Mat. Mekh., 70:2 (2006), 236–249 (Russian)  mathscinet  zmath
    [28] Mukharlyamov, R. G., “Differential-Algebraic Equations of Programmed Motions of Lagrangian Dynamical Systems”, Mech. Solids, 46:4 (2011), 534–543  crossref  adsnasa  elib; Izv. Akad. Nauk. Mekh. Tverd. Tela, 2011, no. 4, 50–61 (Russian)
    [29] Goryachev, D. N., Certain General Integrals in the Problem on the Motion of a Rigid Body, Warsaw, 1910, 62 pp. (Russian)
    [30] Goryachev, D. N., “New Cases of Motion of a Rigid Body around a Fixed Point”, Warshav. Univ. Izv., 3 (1915), 3–14 (Russian)
    [31] Komarov, I. V. and Kuznetsov, V. B., “Generalization of the Goryachev – Chaplygin Gyrostat in Quantum Mechanics”, Zap. Nauchn. Sem. LOMI, 164 (1987), 134–141 (Russian)  mathnet  mathscinet
    [32] Komarov, I. V. and Kuznetsov, V. B., “Semiclassical Quantization of Kowalewski Top”, Theoret. and Math. Phys., 73:3 (1987), 1255–1263  mathnet  crossref  mathscinet  adsnasa; Teoret. Mat. Fiz., 73:3 (1987), 335–347 (Russian)  mathscinet
    [33] Gorr, G. V. and Maznev, A. V., “Integration of the Rigid Body Dynamics on the Invariant Manifold”, Mekh. Tverd. Tela, 46 (2016), 25–36 (Russian)  mathscinet
    [34] Koshlyakov, V. N., Rodrigues – Hamiltion Parameters and Their Applications in Rigid Body Mechanics, Institute of Mathematics NAS of Ukraine, Kiev, 1994, 176 pp. (Russian)  mathscinet
    [35] Lurie, A. I., Analytical Mechanics, Springer, Berlin, 2002, IV, 864 pp.  mathscinet  zmath
    [36] Kharlamov, P. V., “Kinematic Interpretation of the Motion of a Body with a Fixed Point”, J. Appl. Math. Mech., 28:3 (1964), 615–621  crossref  mathscinet  zmath; Prikl. Mat. Mekh., 28:3 (1964), 502–507 (Russian)  zmath
    [37] Gorr, G. V. and Nosyreva, E. P., “Behavior of the Immovable Hodograph for Asymptotically Permanent Motions of the Gyrostat in the Generalized Problem of Dynamics”, Mekh. Tverd. Tela, 26 (1994), 13–20 (Russian)  mathscinet  zmath



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