Geodesic
On asymptotic motions of a heavy rigid body in the Bobylev–Steklov case
Russian journal of nonlinear dynamics, Tome 12 (2016) no. 4, pp. 651-661.

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The Bobylev–Steklov solution belongs to one of the most well-known particular solutions of the Euler–Poisson equation of the problem of motion of a heavy rigid body with a fixed point. It is characterized by two linear invariant relations and can be expressed as elliptic functions of time. The interpretation of the motion of the Bobylev–Steklov gyroscope was carried out by P.V. Kharlamov using the Poinsot method. Analysis of the neighborhood of the Bobylev–Steklov solution in the integral manifold of the Euler–Poisson equations was presented by B.S. Bardin for the case where this solution describes pendulum motions. It is therefore of interest to study the general case of the above-mentioned manifold. Using the first Lyapunov method, a new class of asymptotic motions is obtained for a heavy rigid body whose limit motions are described by the Bobylev–Steklov solution.
Keywords: the first Lyapunov method
Mots-clés : Bobylev–Steklov solution. 
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G. V. Gorr. On asymptotic motions of a heavy rigid body in the Bobylev–Steklov case. Russian journal of nonlinear dynamics, Tome 12 (2016) no. 4, pp. 651-661. http://geodesic.mathdoc.fr/item/ND_2016_12_4_a7/

[1] Kneser A., “Studien über Bewegungsvorgänge in der Umgebung instabiler Gleichgewichtsglage (zwieter Aufsatz)”, J. Reine Angew. Math., 11:3 (1897), 186–223 | MR

[2] Perron O., “Über Stabilität und asymptotischen Verhalten der Integrale von Differentialgleichungssystem”, Math. Z., 29 (1929), 129–160 | DOI | MR

[3] Bohl P., “Über die Bewegung eines mechanischen Systems in der Nähe einer Gleichgewichtslage”, J. Reine Angew. Math., 127:3 (1904), 179–276 | MR | Zbl

[4] Cesari L., Asymptotic behavior and stability problems in ordinary differential equations, Ergeb. Math. Grenzgeb. (N. F.), 16, 2nd ed., Springer, Berlin, 1963, viii+271 pp. | MR | MR | Zbl

[5] Markeev A. P., Linear Hamiltonian systems and some problems of stability of the satellite center of mass, R Dynamics, Institute of Computer Science, Izhevsk, 2009 (Russian)

[6] Bolotin S. V., Kozlov V. V., “Asymptotic solutions of the equations of dynamics”, Mosc. Univ. Mech. Bull., 35:3–4 (1980), 82–88 | Zbl | Zbl

[7] Kozlov V. V., Palamodov V. P., “Asymptotic solutions of equations of classical mechanics”, Sov. Math. Dokl., 25 (1982), 335–339 | MR | Zbl

[8] Markeev A. P., “Plane and quasi-plane rotations of a heavy rigid body about a fixed point”, Izv. AN SSSR. Mekh. Tverd. Tela, 1988, no. 4, 29–36 (Russian)

[9] Markeev A. P., “The stability of the plane motions of a rigid body in the Kovalevskaya case”, J. Appl. Math. Mech., 65:1 (2001), 47–54 | DOI | MR

[10] Markeev A. P., “On the stability of the regular precession of an asymmetric gyroscope”, Dokl. Phys., 47:11 (2002), 833–837 | DOI | MR

[11] Markeyev A. P., “The stability of the Grioli precession”, J. Appl. Math. Mech., 67:4 (2003), 497–510 | DOI | MR | Zbl

[12] Markeyev A. P., “The pendulum-like motions of a rigid body in the Goryachev – Chaplygin case”, J. Appl. Math. Mech., 68:2 (2004), 249–258 | DOI | MR | Zbl

[13] Lyapunov A. M., The general problem of the stability of motion, Gostekhizdat, Moscow, 1950, 471 pp. (Russian) | MR

[14] Varkhalev Yu. P., Gorr G. V., “Asymptotically pendulum-like motions of the Hess – Appel'rot gyroscope”, J. Appl. Math. Mech., 48:3 (1984), 353–356 | DOI

[15] Bryum A. Z., “Investigation of the regular precession of a heavy rigid body with a fixed point by Lyapunov's first method”, Mekh. Tverd. Tela, 1987, no. 19, 68–72 (Russian) | Zbl

[16] Mettler E., “Periodische und asymptotische Bewegungen des unsymmetrischen schweren Kreisels”, Math. Z., 43:1 (1937), 59–100 | DOI | MR

[17] Varkhalev Yu. N., Kovalev V. M., “Asymptotically steady motions of the gyrostat relative to the inclined axis”, Mekh. Tverd. Tela, 1990, no. 22, 43–48 (Russian) | MR | Zbl

[18] Gorr G. V., Kovalev A. I., Dynamics of a gyrostat, Naukova Dumka, Kiev, 2013

[19] Varkhalev Yu. N., Gorr G. V., “The first Lyapunov method in the study of asymptotic motions in rigid body dynamics”, Mekh. Tverd. Tela, 1992, no. 24, 25–41 | MR | Zbl

[20] Bobylev D. N., “On a particular solution of the differential equations of a heavy rigid body rotation around of a fixed point”, Tr. Otdel. Fiz. Nauk Obsch. Lyubit. Estestvozn., 8:2 (1896), 21–25 (Russian)

[21] Steklov V. A., “A certain case of motion of a heavy rigid body having a fixed point”, Tr. Otdel. Fiz. Nauk Obsch. Lyubit. Estestvozn., 8:2 (1896), 19–21 (Russian)

[22] Kharlamov P. V., Lectures on the dynamics of a rigid body, NGU, Novosibirsk, 1965 (Russian)

[23] Bardin B. S., “On stability orbital stability of pendulum like motions of a rigid body in the Bobylev – Steklov case”, Nelin. Dinam., 5:4 (2009), 535–550 (Russian)

[24] Malkin I. G., Theory of stability of motion, Univ. of Michigan, Ann Arbor, Mich., 1958, 472 pp. | MR

[25] Borisov A. V., Mamaev I. S., Rigid body dynamics: Hamiltonian methods, integrability, chaos, R Dynamics, Institute of Computer Science, Izhevsk, 2005 (Russian) | MR