Geodesic
Estimates of Solutions During Motion
Russian journal of nonlinear dynamics, Tome 16 (2020) no. 3, pp. 517-525.

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

This paper presents secure upper and lower estimates for solutions to the equations of rigid body motion in the Euler case (in the absence of external torques). These estimates are expressed by simple formulae in terms of elementary functions and are used for solutions that are obtained in a neighborhood of the unstable steady rotation of the body about its middle axis of inertia. The estimates obtained are applied for a rigorous explanation of the flip-over phenomenon which arises in the experiment with Dzhanibekov’s nut.
Keywords: permanent (steady) rotation, middle axis of inertia, estimates of solutions to differential equations.
Mots-clés : Euler top 
@article{ND_2020_16_3_a7,
     author = {V. F. Zhuravlev and G. M. Rozenblat},
     title = {Estimates of {Solutions} {During} {Motion}},
     journal = {Russian journal of nonlinear dynamics},
     pages = {517--525},
     publisher = {mathdoc},
     volume = {16},
     number = {3},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ND_2020_16_3_a7/}
}
TY  - JOUR
AU  - V. F. Zhuravlev
AU  - G. M. Rozenblat
TI  - Estimates of Solutions During Motion
JO  - Russian journal of nonlinear dynamics
PY  - 2020
SP  - 517
EP  - 525
VL  - 16
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ND_2020_16_3_a7/
LA  - en
ID  - ND_2020_16_3_a7
ER  - 
%0 Journal Article
%A V. F. Zhuravlev
%A G. M. Rozenblat
%T Estimates of Solutions During Motion
%J Russian journal of nonlinear dynamics
%D 2020
%P 517-525
%V 16
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ND_2020_16_3_a7/
%G en
%F ND_2020_16_3_a7
V. F. Zhuravlev; G. M. Rozenblat. Estimates of Solutions During Motion. Russian journal of nonlinear dynamics, Tome 16 (2020) no. 3, pp. 517-525. http://geodesic.mathdoc.fr/item/ND_2020_16_3_a7/

[1] Arkhangel'skii, Yu. A., Analytical Dynamics of a Rigid Body, Nauka, Moscow, 1977, 328 pp. (Russian)

[2] Appel, P., Traité de Mécanique rationnelle, v. 2, Dynamique des systèmes. Mécanique analytique, 6th ed., Gauthier-Villars, Paris, 1953, 584 pp. | MR

[3] Dokl. Akad. Nauk, 451:4 (2013), 399–403 (Russian) | DOI | MR

[4] Zhuravlev, V. F., Petrov, A. G., and Shunderyuk, M. M., Selected Problems of Hamiltonian Mechanics, Lenand, Moscow, 2015, 304 pp. (Russian)

[5] Dokl. Akad. Nauk, 485:2 (2019), 176–181 (Russian) | DOI

[6] Rozenblat, G. M., Gyroscopic Effects in Mechanics of Solids, 2nd ed., Lenand, Moscow, 2020, 128 pp. (Russian)

[7] Hadamard, J., “Sur la précession dans le mouvement d'un corps pesant de révolution fixé par un point de son axe”, Bull. Sci. Math., 19 (1895), 228–230

[8] MacMillan, W. D., Dynamics of Rigid Bodies, McGraw-Hill, New York, 1936, 478 pp. | MR | Zbl

[9] Zhuravlev, V. F., “On the Question of Estimates of the Magnus Effect”, Dokl. Akad. Nauk SSSR, 226:3 (1976), 541–543 (Russian) | MR | Zbl

[10] Zhuravlev, V. F., Foundations of Theoretical Mechanics, 3rd ed., Fizmatlit, Moscow, 2008, 304 pp. (Russian) | MR

[11] 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

[12] Magnus, K., Kreisel: Theorie und Anwendungen, Springer, Berlin, 1971, XII + 493 pp. | MR | Zbl

[13] Byrd, P. F. and Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Scientists, Grundlehren Math. Wiss., 67, 2nd ed., rev., Springer, Heidelberg, 1971, xvi+358 pp. | MR | Zbl

[14] Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G., Higher Transcendental Functions: Vol. 3. Based, in Part, on Notes Left by Harry Bateman, McGraw-Hill, New York, 1955, xvii+292 pp. | MR

[15] Ashbaugh, M. S., Chicone, C. C., and Cushman, R. H., “The Twisting Tennis Racket”, J. Dynam. Differential Equations, 3:1 (1991), 67–85 | DOI | MR | Zbl