Cinematic Interpretation of Motion a Rigid Body in a new Solution of Grioli Equations

Capa

Citar

Texto integral

Acesso aberto Acesso aberto
Acesso é fechado Acesso está concedido
Acesso é fechado Somente assinantes

Resumo

In the article, a new solution is obtained for the problem on motion of a rigid body, having a fixed point, under the action of potential and gyroscopic forces. With use of the modified Poisson method, proposed by the author, it is shown that the motion of the body in this solution can be presented by rolling without sliding of the ellipsoid of inertia of the body along a plane fixed in the immovable space. This result may be considered as an analysis of Poisson result on interpretation of motion a rigid body in Euler solution.

Sobre autores

G. Gorr

Steklov Mathematical Institute RAS

Autor responsável pela correspondência
Email: gvgorr@gmail.com
Russia, Moscow

Bibliografia

  1. Poinsot L. Thèorie nouvelle de la rotation des corps // J. Math. Pures et Appl., 1851, Bd. 1, no. 16, pp. 289–336.
  2. Sylvester J.J. On the motion of a rigid body acted on by no external forces // Philos. Trans. Roy. Soc. London, 1866, vol. 156, pp. 757–780.
  3. Mac-Cullagh J. On the rotation of a solid body // Proc. Roy. Irish Acad., 1840–1844, vol. 2, pp. 542–545; 1845–1847, vol. 3, pp. 370–371.
  4. Darboux G. Sur la theorie de Poinsot et sur des mouvements correspondants a la meme polhodie // C. R. Acad. Sci., 1885, vol. 101, pp. 1555–1561.
  5. Darboux G. Sur le mouvement d’un corps pesant de revolution fixe par un point de son axe // J. Math. Pures et Appl., 1885, vol. 1, pp. 403–430.
  6. Jacobi C.G.J. Sur la rotation d’un corps de rèvolution grave autour d’un point quelconque de son axe // in: Gesammelte Werke. Berlin: G. Reimer, 1882. B. 2, pp. 493–510.
  7. Hess W. Über das Problem der Rotation // Math. Ann., 1882, vol. 20, pp. 461–470.
  8. Hess W. Ūber des Jacobische Theorem von der Ersetzbarkeit einer Lagrangeschen Rotation durch zwei Poinsotische Rotation // Z. Math. Phys., 1888, vol. 33, pp. 292–305.
  9. Zhukovskii N.E. Geometric interpretation of the case considered by Kovalevskaya of the motion of a heavy rigid body about a fixed point // in: Coll. Sci. Papers in 7 Vols. Moscow; Leningrad: Gostekhizdat, 1948, vol. 1. pp. 294–339. (in Russian)
  10. Suslov G.K. Theoretical Mechanics. Moscow: Gostekhizdat,, 1946. 655 p. (in Russian)
  11. Gorr G.V., Kudryashova L.V., Stepanova L.A. Classical Problems on Dynamics of Rigid Body. Kiev: Naukova Dumka, 1978. 296 p. (in Russian)
  12. Gashenenko I.N., Gorr G.V., Kovalev A.M. Classical Problems on Dynamics of Rigid Body. Kyiv: Naukova Dumka, 2012. 402 с. (in Russian)
  13. Kharlamov P.V. Kinematic interpretation of the motion of a body with a fixed point // JAMM, 1964, vol. 28 (3), pp. 502–507.
  14. Gorr G.V. About one approach to the application of Poinsot theorem of kinematic interpretation of the motion of a body with a fixed point // Rigid Body Mech., vol. 42 (Inst. Prikl. Mat. Mekh., Donetsk, 2012), pp. 26–36.
  15. Gorr G.V. On one analogue of Poinsot interpretation of Euler solutionin the problem of rigid body motion in the potential force field // JAMM, 2020, vol. 84 (1), pp. 13–25.
  16. Gorr G.V. On three invariant relations of the equations of motion of a body in a potential field of force // JAMM, 2020, vol. 84 (1), pp. 20–32.
  17. Grioli G. Questioni di dinamica del corpo rigido // Atti. Accad. Naz. Lincei, Rend. Cl. Sci. Fis., Mat. e Natur., 1963, vol. 35, f. 1–2, pp. 35–39.
  18. Yehia H.M. New generalizations of all the known integrable problems in rigid-body dynamics // J. Phys. A.: Math. Gen., 1999, vol. 32, pp. 7565–7580.
  19. Yehia H.M. Equivalent mechanical systems with cyclic coordinates and new in-tegrable problems // Int. J. Non-Linear Mech., 2001, vol. 36, pp. 89–105.
  20. Gorr G.V. On a class of solutions of the dynamics equations for a rigid body acted upon by the potential and gyroscopic forces // JAMM, 2018, vol. 82 (5), pp. 547–558.
  21. Levi-Civita T., Amaldi W. Course in Theoretical Mechanics in 2 Vol. Moscow: Inostr. Lit., 1951. 555 p. (in Russian)
  22. Gorr G.V. Invariant Relations of Equations of Rigid Body Dynamics (Theory, Results, Comments). Moscow; Izhevsk: Inst. Comput. Res., 2017. 424 p. (in Russian)
  23. Kharlamov M.P. Symmetry in systems with gyroscopic forces // Rigid Body Mech., 1983, vol. 15, pp. 87–93.
  24. Gashenenko I.N. Poinsot kinematic representation of the motion of a body in the Hess case // Rigid Body Mech., vol. 40 (Inst. Prikl. Mat. Mekh., Donetsk, 2010), pp. 12–20.
  25. Markeev A.P. On Poinsot geometric interpretation of a rigid body motion in Euler case // Probl. Mech. Control. Motion. Nonlin. Dyn. Syst. Intercollege collection of scientific works. Perm’, 1981, pp. 123–131.
  26. Gorr G.V. An approach in studying gyrostat motion with variable gyrostatic moment // Vestn. Udmurt. Univ. Matem. Mekh. Komp’yut. Nauki., 2021, vol. 31, f. 1, pp. 1–14. (in Russian)
  27. Goryachev D.N. New integrability cases for Euler dynamical equations // Warsaw Univ. Proc., 1916, vol. 3, pp. 1–13. (in Russian)
  28. Goryachev D.N. New cases of motion of a rigid body about a fixed point // Warsaw Univ. Proc., 1915, vol. 3, pp. 1–11. (in Russian)
  29. Yehia H.M. New integrable problems in the dynamics of rigid bodies with Kovalevskaya configuration. I – The case of axisymmetric forces // Mech. Res. Com., 1996, vol. 23, no. 5, pp. 423–437.
  30. Borisov A.V., Mamaev I.S. Dynamics of a Rigid Body. Izhevsk: Sci. & Pub. Center. R&C Dyn., 2001. 84 p. (in Russian)
  31. Komarov I.V., Kuznetsov V.B. Generalized Goryachev–Chaplygin gyrostate in the quantum mechanics // Dif. Geom., Lie Groups&Mech. Proc. Sci. Sem. LOMI NAS USSR, 1987, vol. 9, pp. 134–141. (in Russian)
  32. Komarov I.V., Kuznetsov V.B. Quasi-classical quantization of Kovalevskaya top // Theor. & Math. Phys., 1987, vol. 73 (3), pp. 335–347. (in Russian)
  33. Gorr G.V. A Complex approach to the interpretation of the motion of a solid with a fixed point // Mech. Solids, 2021, vol. 56, no. 6, pp. 932–946. (in Russian)

Declaração de direitos autorais © Г.В. Горр, 2023

Este site utiliza cookies

Ao continuar usando nosso site, você concorda com o procedimento de cookies que mantêm o site funcionando normalmente.

Informação sobre cookies