Geodesic
On the Dynamics of a Rigid Body in the Hess Case at High-Frequency Vibrations of a Suspension Point
Russian journal of nonlinear dynamics, Tome 16 (2020) no. 1, pp. 59-84.

Voir la notice de l'article provenant de la source Math-Net.Ru

The motion of a heavy rigid body with a mass geometry corresponding to the Hess case is considered. The suspension point of the body is assumed to perform high-frequency periodic vibrations of small amplitude in the three-dimensional space. It is proved that for any law of vibrations of this type, the approximate autonomous equations of the body motion admit an invariant relation (the first integral at the zero level), which coincides with a similar relation that exists in the Hess case of the motion of a body with a fixed point. In the approximate equations of motion written in Hamiltonian form, the cyclic coordinate is introduced and the corresponding reduction is performed. For the laws of vibration of the suspension point corresponding to the integrable cases (when there is another cyclic coordinate in the system), a detailed study of the model one-degree-of-freedom system is given. For the nonintegrable cases, an analogy with the approximate problem of the motion of a Lagrange top with a vibrating suspension point is drawn, and the results obtained earlier for the top are used. Some properties of the body motion at the nonzero level of the above invariant relation are also discussed.
Keywords: Hess case, high-frequency vibrations, integrable case, reduced system
Mots-clés : Lagrange top. 
@article{ND_2020_16_1_a5,
     author = {O. V. Kholostova},
     title = {On the {Dynamics} of a {Rigid} {Body} in the {Hess} {Case} at {High-Frequency} {Vibrations} of a {Suspension} {Point}},
     journal = {Russian journal of nonlinear dynamics},
     pages = {59--84},
     publisher = {mathdoc},
     volume = {16},
     number = {1},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ND_2020_16_1_a5/}
}
TY  - JOUR
AU  - O. V. Kholostova
TI  - On the Dynamics of a Rigid Body in the Hess Case at High-Frequency Vibrations of a Suspension Point
JO  - Russian journal of nonlinear dynamics
PY  - 2020
SP  - 59
EP  - 84
VL  - 16
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ND_2020_16_1_a5/
LA  - en
ID  - ND_2020_16_1_a5
ER  - 
%0 Journal Article
%A O. V. Kholostova
%T On the Dynamics of a Rigid Body in the Hess Case at High-Frequency Vibrations of a Suspension Point
%J Russian journal of nonlinear dynamics
%D 2020
%P 59-84
%V 16
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ND_2020_16_1_a5/
%G en
%F ND_2020_16_1_a5
O. V. Kholostova. On the Dynamics of a Rigid Body in the Hess Case at High-Frequency Vibrations of a Suspension Point. Russian journal of nonlinear dynamics, Tome 16 (2020) no. 1, pp. 59-84. http://geodesic.mathdoc.fr/item/ND_2020_16_1_a5/

[1] Hess, W., “Über die Eulerschen Bewegungsgleichungen und eine neue particuläre Lösung des Problems der Bewegung eines starren schweren Körpers um einen festen Punkt”, Math. Ann., 37:2 (1890), 178–180 | DOI | MR

[2] Appel'rot, G. G., “Concerning Section 1 of the Memoir of S. V. Kovalevskaya «Sur le problème de la rotation d'un corps solide autour d'un point fixe», and the Appendix to This Paper”, Mat. Sb., 16:3 (1892), 483–507 (Russian)

[3] Zhukovsky, N. E., “Hess' Loxodromic Pendulum”, Collected Works, v. 1, Gostekhizdat, Moscow, 1937, 332–348 (Russian)

[4] Nekrassov, P. A., “Étude analytique d'un cas du mouvement d'un corps pesant autour d'un point fixe”, Mat. Sb., 18:2 (1896), 161–274 (Russian) | MR

[5] Golubev, V. V., Lectures on Integration of the Equations of Motion of a Rigid Body about a Fixed Point, Israel Program for Scientific Translations, Jerusalem, 1960, 229 pp. | MR | Zbl

[6] Borisov, A. V. and Mamaev, I. S., Rigid Body Dynamics: Hamiltonian Methods, Integrability, Chaos, R Dynamics, Institute of Computer Science, Izhevsk, 2005, 576 pp. (Russian) | MR

[7] Sretenskii, L. N., “Some Integrability Cases for the Equations of Gyrostat Motion”, Dokl. Akad. Nauk SSSR, 149:2 (1963), 292–294 (Russian) | MR | Zbl

[8] Burov, A. A., “On a Partial Integral in the Problem of the Motion of a Heavy Rigid Body Suspended on a Rod”, Research Problems of Stability and Stabilisation of Motion, Computing Centre of the USSR Acad. Sci., Moscow, 1986, 93–95 (Russian) | MR

[9] Burov, A. A., “On Partial Integrals of the Equations of Motion of a Rigid Body on a Frictionless Horizontal Plane”, Research Problems of Stability and Stabilisation of Motion, eds. V. V. Rumyantsev, V. S. Sergeev, S. Ya. Stepanov, A. S. Sumbatov, Computing Centre of the USSR Acad. Sci., Moscow, 1985, 118–121 (Russian) | MR

[10] Chaplygin, S. A., “Some Cases of Motion of a Rigid Body in a Fluid: 1, 2”, Collected Works, v. 1, Gostekhizdat, Moscow, 1948, 136–311 (Russian) | MR

[11] Dokl. Akad. Nauk SSSR, 266:6 (1982), 1298–1300 (Russian) | MR | Zbl

[12] Prikl. Mat. Mekh., 48:5 (1984), 745–749 (Russian) | DOI | MR

[13] Prikl. Mat. Mekh., 67:2 (2003), 256–265 (Russian) | DOI | MR | Zbl

[14] Tr. Mat. Inst. Steklova, 294 (2016), 268–292 (Russian) | DOI | MR | Zbl

[15] Lubowiecki, P. and Żoła̧dek, H., “The Hess – Appelrot System: 1. Invariant Torus and Its Normal Hyperbolicity”, J. Geom. Mech., 4:4 (2012), 443–467 | DOI | MR | Zbl

[16] Lubowiecki, P. and Żoła̧dek, H., “The Hess – Appelrot System: 2. Perturbation and Limit Cycles”, J. Differential Equations, 252:2 (2012), 1701–1722 | DOI | MR | Zbl

[17] Kurek, R., Lubowiecki, P., and Żoła̧dek, H., “The Hess – Appelrot System: 3. Splitting of Separatrices and Chaos”, Discrete Contin. Dyn. Syst. A, 38:4 (2018), 1955–1981 | DOI | MR | Zbl

[18] Żoła̧dek, H., “Perturbations of the Hess – Appelrot and the Lagrange Cases in the Rigid Body Dynamics”, J. Geom. Phys., 142 (2019), 121–136 | DOI | MR

[19] Bizyaev, I. A., Borisov, A. V., and Mamaev, I. S., “The Hess – Appelrot Case and Quantization of the Rotation Number”, Regul. Chaotic Dyn., 22:2 (2017), 180–196 | DOI | MR | Zbl

[20] Dokl. Akad. Nauk, 427:6 (2009), 771–775 (Russian) | DOI | MR | Zbl

[21] Prikl. Mat. Mekh., 75:2 (2011), 193–203 (Russian) | DOI | MR | Zbl

[22] Kholostova, O. V., “On a Case of Periodic Motions of the Lagrangian Top with Vibrating Fixed Point”, Regul. Chaotic Dyn., 4:4 (1999), 81–93 | DOI | MR | Zbl

[23] Dokl. Ross. Akad. Nauk, 375:5 (2000), 627–630 (Russian) | DOI | MR

[24] Prikl. Mat. Mekh., 64:5 (2000), 858–868 (Russian) | DOI | MR | Zbl

[25] Izv. Akad. Nauk. Mekh. Tverd. Tela, 2012, no. 4, 3–10 (Russian) | DOI

[26] Belichenko, M. V. and Kholostova, O. V., “On the Stability of Stationary Rotations in the Approximate Problem of Motion of Lagrange's Top with a Vibrating Suspension Point”, Nelin. Dinam., 13:1 (2017), 81–104 (Russian) | DOI | MR | Zbl

[27] Belichenko, M. V., “On the Stability of Pendulum-Type Motions in the Approximate Problem of Dynamics of a Lagrange Top with a Vibrating Suspension Point”, Russian J. Nonlinear Dyn., 14:2 (2018), 243–263 | MR | Zbl

[28] Gradshtein, I. S. and Ryzhik, I. M., Table of Integrals, Series, and Products, 7th ed., Acad. Press, Amsterdam, 2007, 1200 pp. | MR | Zbl

[29] Malkin, I. G., Theory of Stability of Motion, U.S. Atomic Energy Commission, Washington, D.C., 1952, 456 pp.

[30] Byrd, P. F. and Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Scientists, Grundlehren Math. Wiss., 67, 2nd ed., rev., Springer, Berlin, 1971, XVI, 360 pp. | MR | Zbl

[31] Karapetyan, A. V. and Rumyantsev, V. V., Stability of Conservative and Dissipative Systems, Itogi Nauki Tekh. Ser. Obshch. Mekh., 6, VINITI, Moscow, 1983, 132 pp. (Russian) | MR

[32] Karapetyan, A. V., The Stability of Steady Motions, Editorial URSS, Moscow, 1998, 168 pp. (Russian)

[33] Staude, O., “Über permanente Rotationsaxen bei der Bewegung eines schweren Körpers um einen festen Punkt”, J. Reine Angew. Math., 113:4 (1894), 318–334 | MR

[34] Kholostova O. V., Stability Investigation of Staude's Permanent Rotations, R Dynamics, Institute of Computer Science, Izhevsk, 2008 (Russian)