On a pendulum motion in multi-dimensional space. Part 1. Dynamical systems

M. V. Shamolin

Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, no. 3 (2017), pp. 41-64

Résumé

In the proposed cycle of work, we study the equations of the 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 the 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 thit work, we derive the general multi-dimensional dynamic equations of the systems under study.

Détail

Keywords: multi-dimensional rigid body, non-conservative force field, dynamical system, case of integrability.

Bibliographie

[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)

[3] Shamolin M. V., “Variety of cases of integrability in dynamics of lower-, and multi-dimensional body in nonconservative field”, Dinamicheskie sistemy, Itogi nauki i tekhniki. Ser.: “Sovremennaia matematika i ee prilozheniia. Tematicheskie obzory”, 125, 2013, 5–254 (in Russina)

[4] Pokhodnya N. V., Shamolin M. V., “Some cases of integrability of dynamic systems in transcedent functions”, Vestnik of Samara State University. Natural Science Series, 2013, no. 9/1(110), 35–41 (in Russian)

[5] Shamolin M. V., “Variety of types of phase portraits in dynamics of a rigid body interacting with a resisting medium”, Physics Doklady, 349:2 (1996), 193–197 (in Russian)

[6] Shamolin M. V., “With Variable Dissipation: Approaches, Methods, and Applications”, Journal of Mathematical Sciences, 14:3 (2008), 3–237 (in Russian)

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

[8] Trofimov V. V., “Symplectic structures on symmetruc spaces automorphysm groups”, Moscow University Mathematics Bulletin, 1984, no. 6, 31–33 (in Russian)

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

[10] Shamolin M. V., Methods of analysis of various dissipation dynamical systems in dynamics of a rigid body, Izd-vo “Ekzamen”, M., 2007, 352 pp. (in Russian)

[11] Shamolin M. V., “Some model problems of dynamics for a rigid body interacting with a medium”, International Applied Mechanics, 43:10 (2007), 49–67 (in Russian)

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

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

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