CASES OF INTEGRABILITY CORRESPONDING TO THE PENDULUM MOTION IN THREE-DIMENSIONAL SPACE



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Abstract

In this article, we systemize some results on the study of the equations of spatial motion of dynamically symmetric fixed 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 a spatial motion of a free 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. The obtained results are systematized and served in the invariant form. We also show the nontrivial topological and mechanical analogies.

About the authors

M. V. Shamolin

Institute of Mechanics, Lomonosov Moscow State University, Moscow, 119192, Russian Federation.

Author for correspondence.
Email: morenov.sv@ssau.ru

References

  1. Shamolin M.V. Cases of integrability corresponding to the pendulum motion on the plane, Vestnik Samarskogo gosudarstvennogo universiteta. Ser.: Esyestvennonauchnaya , 2015, no. 10(132), pp. 91–113. (in Russ.)
  2. Shamolin M.V. New integrable cases and families of portraits in the plane and spatial dynamics of a rigid body interacting with a medium. Journal of Mathematical Sciences, 2003, 114, no. 1, pp. 919–975 .
  3. Shamolin M.V. Variety of cases of integrability in dynamics of lower-, and multidimensional body in nonconservative field. Itogi nauki i tekhniki. Ser. Sovremennaya matematika i ee prolodzeniya. Tematicheskie obzory , 125, Dynamical Systems, 2013, pp. 5–254 .
  4. Pokhodnya N.V., Shamolin M.V. New case of integrability in dynamics of multi-dimensional body. Vestnik Samarskogo gosudarstvennogo universiteta. Ser.: Esyestvennonauchnaya , 2012, no. 9/1(110), pp. 35–41 .
  5. Shamolin M.V. Variety of types of phase portraits in dynamics of a rigid body interacting with a resisting medium. Doklady RAN , 1996, 349, no. 2, pp. 193–197 .
  6. Shamolin M.V. Dynamical Systems With Variable Dissipation: Approaches, Methods, and Applications. Fund. Prikl. Mat. , 2008, 14, no. 3, pp. 3–237. .
  7. Arnold V.I., Kozlov V.V., Neyshtadt A.I. Mathematical aspect in classical and celestial mechanics. M.: VINITI, 1985, 304 p. .
  8. Trofimov V.V. Symplectic structures on symmetruc spaces automorphysm groups. Vestnik Moskovskogo Universiteta. Ser. 1. Matematika. Mekhanika , 1984, no. 6, pp. 31–33 .
  9. Trofimov V.V., Shamolin M.V. Geometrical and dynamical invariants of integrable Hamiltonian and dissipative systems. Fund. Prikl. Mat. , 2010, 16, no. 4, pp. 3–229 .
  10. Shamolin M.V. Methods of analysis of various dissipation dynamical systems in dynamics of a rigid body. M.: Ekzamen, 2007, 352 pp. .
  11. Shamolin M.V. Classes of variable dissipation systems with nonzero mean in the dynamics of a rigid body. Journal of Mathematical Sciences, 2004, 122, no. 1, pp. 2841–2915 .
  12. Shamolin M.V. Some model problems of dynamics for a rigid body interacting with a medium. Prikl. Mekh. , 2007, 43, no. 10, pp. 49–67.
  13. Shamolin M.V. New integrable cases in dynamics of a medium-interacting body with allowance for dependence of resistance force moment on angular velocity. Prikl. Mat. Mekh. , 2008, 72, no. 2, pp. 273–287 .
  14. Shamolin M.V. Integrability of some classes of dynamic systems in terms of elementary functions. Vestnik Moskovskogo Universiteta. Ser. 1. Matematika. Mekhanika , no. 3, pp. 43–49 (2008) .
  15. Shamolin M.V. Stability of a rigid body translating in a resisting medium. Prikl. Mekh. , 2009, 45, no. 6, pp. 125–140 .

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