Geodesic
On orbital stability pendulum-like oscillations and rotation of symmetric rigid body with a fixed point
Russian journal of nonlinear dynamics, Tome 8 (2012) no. 2, pp. 249-266.

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

We deal with the problem of orbital stability of planar periodic motions of a heavy rigid body with a fixed point. We suppose that the mass center of the body is located in the equatorial plane of the inertia ellipsoid. Unperturbed motions represent oscillations or rotations of the body around a principal axis, keeping a fixed horizontal position. Local coordinates are introduced in a neighborhood of the unperturbed periodic motion and equations of perturbed motion are obtained in Hamiltonian form. Domains of orbital instability are established by means of linear analysis. Outside of the above domains nonlinear study is performed. The nonlinear stability problem is reduced to a stability problem of a fixed point of symplectic map generated by the equations of perturbed motion. Coefficients of the above map are obtained numerically. By analyzing of the coefficients mentioned rigorous results on orbital stability or instability are obtained. In the case of oscillations with small amplitudes as well as in the case of rotations with high angular velocities the problem of orbital stability is studied analytically.
Keywords: Hamiltonian system, periodic orbits, normal form, resonance, orbital stability.
Mots-clés : action–angel variables 
@article{ND_2012_8_2_a3,
     author = {B. S. Bardin and A. A. Savin},
     title = {On orbital stability pendulum-like oscillations and rotation of symmetric rigid body with a fixed point},
     journal = {Russian journal of nonlinear dynamics},
     pages = {249--266},
     publisher = {mathdoc},
     volume = {8},
     number = {2},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ND_2012_8_2_a3/}
}
TY  - JOUR
AU  - B. S. Bardin
AU  - A. A. Savin
TI  - On orbital stability pendulum-like oscillations and rotation of symmetric rigid body with a fixed point
JO  - Russian journal of nonlinear dynamics
PY  - 2012
SP  - 249
EP  - 266
VL  - 8
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ND_2012_8_2_a3/
LA  - ru
ID  - ND_2012_8_2_a3
ER  - 
%0 Journal Article
%A B. S. Bardin
%A A. A. Savin
%T On orbital stability pendulum-like oscillations and rotation of symmetric rigid body with a fixed point
%J Russian journal of nonlinear dynamics
%D 2012
%P 249-266
%V 8
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ND_2012_8_2_a3/
%G ru
%F ND_2012_8_2_a3
B. S. Bardin; A. A. Savin. On orbital stability pendulum-like oscillations and rotation of symmetric rigid body with a fixed point. Russian journal of nonlinear dynamics, Tome 8 (2012) no. 2, pp. 249-266. http://geodesic.mathdoc.fr/item/ND_2012_8_2_a3/

[1] Irtegov V. D., “Ob ustoichivosti mayatnikovykh kolebanii giroskopa S. V. Kovalevskoi”, Tr. Kazan. aviats. in-ta, 1968, no. 97, 38–40

[2] Bryum A. Z., “Issledovanie orbitalnoi ustoichivosti pri pomoschi pervykh integralov”, PMM, 53:6 (1989), 873–879 | MR | Zbl

[3] Markeev A. P., “Ob ustoichivosti ploskikh dvizhenii tverdogo tela v sluchae Kovalevskoi”, PMM, 65:1 (2001), 51–58 | MR

[4] Markeev A. P., Medvedev S. V., Chekhovskaya T. N., “K zadache ob ustoichivosti mayatnikovykh dvizhenii tverdogo tela v sluchae Kovalevskoi”, MTT, 2003, no. 1, 3–9

[5] Bardin B. S., “K zadache ob ustoichivosti mayatnikoobraznykh dvizhenii tverdogo tela v sluchae Goryacheva–Chaplygina”, MTT, 2007, no. 2, 14–21

[6] Akhiezer N. I., Elementy teorii ellipticheskikh funktsii, Nauka, M., 1970, 304 pp. | MR | Zbl

[7] Markeev A. P., Tochki libratsii v nebesnoi mekhanike i kosmodinamike, Nauka, M, 1978, 312 pp.

[8] Malkin I. G., Teoriya ustoichivosti dvizheniya, Nauka, M., 1966, 530 pp. | MR | Zbl

[9] Gradshtein I. S., Ryzhik I. M., Tablitsy integralov, summ, ryadov i proizvedenii, Fizmatlit, M., 1963, 1100 pp. | MR

[10] Yakubovich V. Ya., Starzhinskii V. M., Parametricheskii rezonans v lineinykh sistemakh, Nauka, M., 1987, 328 pp. | MR

[11] Markeev A. P., Lineinye gamiltonovy sistemy i nekotorye zadachi ob ustoichivosti dvizheniya sputnika otnositelno tsentra mass, NITs «Regulyarnaya i khaoticheskaya dinamika», Inst. kompyutern. issled., M.–Izhevsk, 2009, 396 pp.

[12] Birkgof Dzh. D., Dinamicheskie sistemy, Izdatelskii dom «Udmurtskii universitet», Izhevsk, 1999, 408 pp. | Zbl

[13] Dzhakalya G. E. O., Metody vozmuschenii dlya nelineinykh sistem, Nauka, M., 1979, 319 pp.

[14] Arnold V. I., “Malye znamenateli i problemy ustoichivosti dvizheniya v klassicheskoi i nebesnoi mekhanike”, UMN, 18:6(114) (1963), 91–192 | MR

[15] Mozer Yu., Lektsii o gamiltonovykh sistemakh, M., Mir, 1973, 167 pp.

[16] Ivanov A. P., Sokolskii A. G., “Ob ustoichivosti neavtonomnoi gamiltonovoi sistemy pri parametricheskom rezonanse osnovnogo tipa”, PMM, 44:6 (1980), 963–970 | MR | Zbl

[17] Markeev A. P., “Ob odnom sposobe issledovaniya ustoichivosti polozhenii ravnovesiya gamiltonovykh sistem”, MTT, 2004, no. 6, 3–12