Geodesic
On the qualitative analysis of a family of differential equations with first integrals of degree more then 2
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 16 (2023) no. 2, pp. 37-48
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We study a family of differential equations that arose as a result of a generalization of the classical integrable cases in rigid body dynamics. The system under study admits polynomial first integrals of the 4th and 6th degrees. Under appropriate constraints on the parameters of the family, the differential equations are interpreted as those of motion of a rigid body in a central force field and an ideal fluid, as well as the equations of motion of an electrically charged body. The qualitative analysis of the equations is done. We find special invariant sets of various dimensions and investigate their Lyapunov stability. For the analysis of the problem, generalizations of the Routh-Lyapunov method and software tools of computer algebra are applied.
Keywords: rigid body, the equations of motion, first integrals, stability, computer algebra.
Mots-clés : invariant sets 
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V. D. Irtegov; T. N. Titorenko. On the qualitative analysis of a family of differential equations with first integrals of degree more then 2. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 16 (2023) no. 2, pp. 37-48. http://geodesic.mathdoc.fr/item/VYURU_2023_16_2_a3/

[1] Oparina E.I., Troshkin O.V., “Stability of Kolmogorov Flow in a Channel with Rigid Walls”, Doklady Physics, 49:10 (2004), 583–587 | DOI | MR

[2] H.M. Yehia, “New Generalizations of the Integrable Problems in Rigid Body Dynamics”, Journal of Physics A: Mathematical and General, 30:20 (1997), 7269–7275 | DOI | MR | Zbl

[3] H.M. Yehia, “New Generalizations of all the Known Integrable Problems in Rigid-Body Dynamics”, Journal of Physics A: Mathematical and General, 32:43 (1999), 7565–7580 | DOI | MR | Zbl

[4] Borisov A.V., Mamaev I.S., Rigid Body Dynamics. Hamiltonian Methods, Integrablity, Chaos, Institute of Computer Science, M., 2005 (in Russian) | MR

[5] E.J. Routh, The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies, MacMillan, London, 1905 | MR

[6] Lyapunov A.M., “On the Permanent Helical Motions of a Rigid Body in Fluid”, Collected Works, v. 1, 1954, 276–319 (in Russian)

[7] Salvadori L., “Sulla Stabilita del Movimento”, Matematiche, 24:1 (1969), 218–239 | MR | Zbl

[8] Irtegov V.D., Titorenko T.N., “About Stationary Movements of the Generalized Kovalevskaya Top and their Stability”, Mechanics of Solids, 54:1 (2019), 81–91 | DOI | DOI

[9] Irtegov V.D., Titorenko T.N., “On the Qualitative Analysis of the Equations of Motion of a Rigid Body in a Magnetic Field”, Journal of Applied and Industrial Mathematics, 16:1 (2022), 58–69 | DOI | MR | Zbl

[10] Yehia H.M., “New Solutions of the Problem on the Motion of a Gyrostat in Potential and Magnetic Fields”, Proceedings of Moscow University. Series: Mathematics and Mechanics, 1985, no. 5, 60–63 (in Russian) | MR

[11] Irtegov V.D., Titorenko T.N., “Computer Algebra Methods in the Study of Nonlinear Differential Systems”, Computational Mathematics and Mathematical Physics, 53:6 (2013), 845–857 | DOI | DOI | MR | Zbl

[12] Irtegov V.D., Titorenko T.N., “The Invariant Manifolds of Systems with First Integrals”, Journal of Applied Mathematics and Mechanics, 73:4 (2009), 379–384 | DOI | MR | Zbl

[13] Irtegov V.D., Titorenko T.N., “Invariant Manifolds in the Clebsch–Tisserand–Brun Problem”, Journal of Applied Mathematics and Mechanics, 76:3 (2012), 268–274 | DOI | MR | Zbl

[14] Lyapunov A.M., The General Problem of the Stability of Motion, Taylor and Francis, London, 1992 | MR | MR | Zbl