Geodesic
On a pendulum motion in multi-dimensional space. Part 3. Dependence of force fields on the tensor of angular velocity
Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 24 (2018) no. 2, pp. 33-54 Cet article a éte moissonné depuis la source Math-Net.Ru

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In the proposed cycle of work, we study the equations of motion of dynamically symmetric fixed $n$-dimensional rigid bodies–pendulums located in a nonconservative force fields. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of a medium. In parallel, we study the problem of motion of a free $n$-dimensional rigid body also located in a similar force fields. Herewith, this free rigid body is influenced by a nonconservative tracing force; under action of this force, either the magnitude of the velocity of some characteristic point of the body remains constant, which means that the system possesses a nonintegrable servo constraint. In this work, we study that case when the force fields linearly depend on the tensor of angular velocity.
Keywords: multi-dimensional rigid body, non-conservative force field, dynamical system, case of integrability. 
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M. V. Shamolin. On a pendulum motion in multi-dimensional space. Part 3. Dependence of force fields on the tensor of angular velocity. Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 24 (2018) no. 2, pp. 33-54. http://geodesic.mathdoc.fr/item/VSGU_2018_24_2_a4/

[1] Shamolin M.V., “Cases of integrability corresponding to the pendulum motion on the plane”, Vestnik of Samara State University. Natural Science Series, 2015, no. 10(132), 91–113 (in Russian)

[2] Shamolin M.V., “Cases of integrability corresponding to the pendulum motion on the three-dimensional space”, Vestnik of Samara State University. Natural Science Series, 2016, no. 3–4, 75–97 (in Russian) | Zbl

[3] Shamolin M.V., “Variety of cases of integrability in dynamics of lower-, and multi-dimensional body in nonconservative field”, Dynamical Systems, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 125, 2013, 5–254 (in Russian)

[4] Pokhodnya N.V., Shamolin M.V., “Certain conditions of integrability of dynamical systems in transcedental functions”, Vestnik of Samara State University. Natural Science Series, 2013, no. 9/1(110), 35–41 (in Russian) | Zbl

[5] Shamolin M.V., “New Cases of Integrable Systems with Dissipation on the Tangent Bundle of a Three-Dimensional Manifold”, Doklady Physics, 477:2 (2017), 168–172 (in Russian) | DOI | MR

[6] Shamolin M.V., “Complete List of First Integrals of Dynamic Equations for a Multidimensional Solid in a Nonconservative Field”, Doklady Physics, 461:5 (2015), 533–536 (in Russian) | DOI | MR

[7] Arnold V.I., Kozlov V.V., Neyshtadt A.I., Mathematical aspect in classical and celestial mechanics, VINITI, M., 1985, 304 pp. (in Russian) | MR

[8] Trofimov V.V., “Symplectic structures on symmetric spaces automorphysm groups”, Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika, 1984, no. 6, 31–33 (in Russian) | Zbl

[9] Trofimov V.V., Shamolin M.V., “Geometrical and dynamical invariants of integrable Hamiltonian and dissipative systems”, Fundamental and Applied Mathematics, 16:4 (2010), 3–229 (in Russian)

[10] Shamolin M.V., “A Multidimensional Pendulum in a Nonconservative Force Field”, Doklady Physics, 460:2 (2015), 165–169 (in Russian) | DOI | MR

[11] Shamolin M.V., “New Case of Integrability in the Dynamics of a Multidimensional Solid in a Nonconservative Field”, Doklady Physics, 453:1 (2013), 46–49 (in Russian) | DOI | MR

[12] Shamolin M.V., “New Cases of Integrable Systems with Dissipation on Tangent Bundles of Two- and Three-Dimensional Spheres”, Doklady Physics, 471:5 (2016), 547–551 (in Russian) | DOI | MR

[13] Shamolin M.V., “New Cases of Integrable Systems with Dissipation on a Tangent Bundle of a Multidimensional Sphere”, Doklady Physics, 474:2 (2017), 177–181 (in Russian) | DOI | MR

[14] Shamolin M.V., “New Cases of Integrable Systems with Dissipation on a Tangent Bundle of a Two-Dimensional Manifold”, Doklady Physics, 475:5 (2017), 519–523 (in Russian) | DOI | MR