Skip to main content
SpringerLink
Log in

Analog of the Poinsot Interpretation of the Euler Solution in the Motion Problem for a Rigid Body in a Potential Field of Forces

  • Published:
Mechanics of Solids Aims and scope Submit manuscript

Abstract

We investigate the motion problem for a rigid body with a fixed point under the action of potential forces. Existence conditions for three invariant relations of the equations of motion are obtained. Dependences of basic variables on time are found. The motion of the ellipsoid of inertia of the body is studied in immovable space. An analog of the Poinsot interpretation of the Euler solution is established: the motion of the body is represented by the ellipsoid of inertia rolling without slip along a plane tangent to it in fixed space.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. S. L. Ziglin, “Branching of solutions and non-existence of first integrals in Hamiltonian systems,” Funkts. Anal. Ego Prilozh. 17 (1), 8–23 (1983).

    MathSciNet  Google Scholar 

  2. S. L. Ziglin, “Separatrices splitting, solutions branching and non-existence of an integral in rigid body dynamics,” Tr. Mosk. Mat. O-va 41, 287–303 (1980).

    MathSciNet  MATH  Google Scholar 

  3. V. V. Kozlov, “On the non-existence of analytic integrals of canonical systems close to integrable ones,” Vestn. Mosk. Univ., Ser. 1: Mat. Mekh., No. 2, 77–82 (1974).

  4. A. V. Borisov, “Necessary and sufficient conditions for integrability of Kirchhoff equations,” Reg. Chaot. Dyn. 1 (2), 61–76 (1996).

    MathSciNet  MATH  Google Scholar 

  5. V. V. Kozlov and D. A. Onishchenko, “Nonintegrability of Kirchhoff’s equations,” Sov. Math. Dokl. 26, 495–498 (1982).

    MATH  Google Scholar 

  6. G. K. Suslov, Theoretical Mechanics (Gostekhizdat, Moscow, 1946) [in Russian].

    Google Scholar 

  7. G. V. Gorr, L. V. Kudryashova, and L. A. Stepanova, Classical Problems of Rigid Body Dynamics (Naukova dumka, Kiev, 1978) [in Russian].

    Google Scholar 

  8. A. V. Borisov and I. S. Mamaev, Rigid Body Dynamics (Research Centre “Regular and chaotic dynamics”, Moscow-Izhevsk, 2001) [in Russian].

  9. G. V. Gorr and A. V. Maznev, Dynamics of a Gyrostat Having a Fixed Point (Donetsk National Univ., Donetsk, 2010) [in Russian].

    MATH  Google Scholar 

  10. L. Poinsot, “Thèorie nouvelle de la rotation des corps,” J. Math. Pures Appl. 1 (16), 289–336 (1851).

    Google Scholar 

  11. L. Euler, “Du mouvement de rotation des corps solides Autour d’un axe variable,” J. Math. Pures Appl. 14, 154–193 (1758, 1765).

  12. J. E. Routh, Dynamics of a System of Rigid Bodies. Part II (Macmillan, London, 1884).

    Google Scholar 

  13. K. Magnus, Theorie und Anwendungen (Springer, Berlin, Heidelberg, 1971).

    MATH  Google Scholar 

  14. A. S. Domogarov, On the Free Motion of a Gyrostat (St. Petersburg, 1893) [in Russian].

    Google Scholar 

  15. P. V. Kharlamov, Lectures on Rigid Body Dynamics (Novosibirsk State Univ., Novosibirsk, 1965) [in Russian].

    Google Scholar 

  16. N. E. Zhukovskii, “On the meaning of geometric interpretation in theoretical mechanics,” in Collection of Scientific Papers in 7 Vols. (Gostekhizdat, Moscow-Leningrad, 1950), Vol. 7, pp. 9–15 [in Russian].

    Google Scholar 

  17. N. E. Zhukovskii, “Hess’s loxodromic pendulum,” in Collection of Scientific Papers in 7 Vols. (Gostekhizdat, Moscow-Leningrad, 1948), Vol. 1, pp. 257–274 [in Russian].

    Google Scholar 

  18. P. V. Kharlamov, “Kinematic interpretation of the motion of a body having a fixed point,” J. Appl. Math. Mech. 28 (3), 615–621 (1964).

    Article  MathSciNet  Google Scholar 

  19. I. N. Gashenenko, G. V. Gorr, and A. M. Kovalev, Classical Problems on Rigid Body Dynamics (Nauk. dumka, Kiev, 2012) [in Russian].

  20. G. V. Gorr and A. M. Kovalev, Motion of a Gyrostat (Naukova dumka, Kiev, 2013) [in Russian].

    Google Scholar 

  21. G. V. Gorr, “On one approach for applying the Poinsot theorem on the kinematic interpretation for the motion of a body with a fixed point,” Mekh. Tverd. Tela, No. 42, 26–36 (2012).

    Google Scholar 

  22. G. V. Gorr and E. K. Shchetinina, “On the motion of a heavy rigid body for two particular cases of S.V. Kovalevskaya’s solution,” Nelineinaya Din. 14 (1), 123–138 (2018).

    Article  MathSciNet  Google Scholar 

  23. G. V. Gorr and A. I. Sinenko, “A kinematic interpretation of the motion of a heavy rigid body with a fixed point,” J. Appl. Math. Mech. 28 (3), 233–241 (2014).

    Article  MathSciNet  Google Scholar 

  24. G. V. Gorr, D. A. Danilyuk, and D. N. Tkachenko, “On kinematic interpretation of body motion for a particular case of D. N. Goryachev–S. A. Chaplygin solution,” Zh. Teor. Prikl. Mekh. (DonNU), No. 3-4 (60), 19–32 (2017).

  25. G. V. Gorr, “On three invariant relations of thee equations of motion of a body in a potential field of force,” Mech. Solids 54 (2), 234–244 (2019).

    Article  ADS  Google Scholar 

  26. T. Levi-Civita and U. Amaldi, Lezioni di Meccanica Razionale. In 2 vol. (Zanichelli, Bologna, 1952), Vol. 2, Part 2.

    MATH  Google Scholar 

  27. P. V. Kharlamov, “On invariant relations of a system of differential equations,” Mekh. Tverd. Tela, No. 6, 15–24 (1974).

    Google Scholar 

  28. G. V. Gorr, Invariant Relations of Equations for the Rigid Body Dynamics (Theory, Results, Comments) (Institute of Computer Science Moscow, Izhevsk, 2017) [in Russian].

    Google Scholar 

  29. A. S. Galiullin, “Inverse dynamics problems of heavy rigid body with one fixed point,” Differ. Uravn. 8 (8), 1357–1362 (1972).

    MathSciNet  MATH  Google Scholar 

  30. D. N. Goryachev, Some General Integrals in the Problem on a Rigid Body Motion (Tip. Varsh. uch. okruga, Warsaw, 1910) [in Russian].

  31. D. N. Goryachev, “New cases of rigid body motion around a fixed point,” Varshav. Univ. Izv., Book 3, 1–11 (1915).

    Google Scholar 

  32. H. M. Yehia, “Transformations of mechanical systems with cyclic coordinates and new integrable problems,” J. Phys. A: Gen. Phys. 341, 11167–11183 (2001).

    Article  ADS  MathSciNet  Google Scholar 

  33. H. M. Yehia, “Kovalevskaya’s integrable case: generalizations and related new results,” Regular Chaotic Dyn. 8 (3), 337–348 (2003).

    Article  ADS  MathSciNet  Google Scholar 

  34. I. V. Komarov and V. B. Kuznetsov, “Generalized Goryachev-Chaplygin gyrostat in the quantum mechanics,” in Differential Geometry, Lie Groups and Mechanics. Notes of the Scientific Seminar in the Leningrad Branch of Steklov Mathematical Institute of the USSR Academy of Sciences (Leningrad, 1987), Vol. 9, pp. 134–141 [in Russian].

    Google Scholar 

  35. I. V. Komarov and V. B. Kuznetsov, “Quasiclassical quantization of the Kovalevskaya top,” Teor. Mat. Fiz. 73 (3), 335–347 (1987).

    Article  MathSciNet  Google Scholar 

  36. G. V. Gorr, A. A. Ilyukhin, A. M. Kovalev, and A. Ya. Savchenko, Nonlinear Analysis of Mechanical Systems Behavior (Naukova Dumka, Kiev, 1984) [in Russian].

    Google Scholar 

  37. F. Klein and A. Sommerfeld, Über die Theorie des Kreisels (Johnson Reprint Corp, New York, 1965).

    MATH  Google Scholar 

  38. G. Grioli, “Esistenza e determinazione delle precessioni regolari dinamicamente possibili per un solido pesante asimmetrico,” Ann. Mat. Pura Appl. 24 (3-4), 271–281 (1947).

    Article  MathSciNet  Google Scholar 

  39. V. A. Steklov, “The new partial solution of differential equations of motion for a rigid body with a fixed point,” Tr. Otd. Fiz. Nauk O-va. Lyubit. Estestvozn. 10 (1), 1–3 (1899).

    Google Scholar 

  40. R. Fabbri, “Sopra una soluzione particolare delle equazioni del moto di un solido pesante ad un punto fisso,” Atti. Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat. 19 (6), 407–415, 495–502, 872–873 (1934).

  41. R. Fabbri, “Sui coni di Poinsot in una particolare rotazione dei solidi pesanti,” Atti. Accad. Naz. Lincei. Rend. Cl. Sci. Fis., Mat. Natur. Ser. 6 19, 872–873 (1934).

    Google Scholar 

  42. E. I. Kharlamova and G. V. Mozalevskaya, Integro-Differential Equation of Rigid Body Dynamics (Naukova dumka, Kiev, 1986) [in Russian].

    MATH  Google Scholar 

  43. E. V. Verkhovod and G. V. Gorr, “Precession-isoconic motions of a rigid body with a fixed point,” J. Appl. Math. Mech. 57 (4), 613–622 (1993).

    Article  MathSciNet  Google Scholar 

  44. E. V. Verkhovod and G. V. Gorr, “New cases of isoconic motions in the generalized problem of dynamics of a rigid body with a fixed point,” J. Appl. Math. Mech. 57 (5), 783–792 (1993).

    Article  MathSciNet  Google Scholar 

  45. I. N. Gashenenko, “Hess case for body motion: Poinsot kinematic presentation,” Mekh. Tverd. Tela, No. 40, 12–20 (2010).

    Google Scholar 

  46. A. Bressan, “Sulle precessini d’un corpo rigido costituenti moti di Hess,” Rend. Semin. Mat. Univ. Padova 27, 276–283 (1957).

    MATH  Google Scholar 

  47. W. Hess, “Über die Euler’schen Bewegungsgleichungen und über eine neue partikuläre Lösung des Problems der Bewegung eines starren schweren Körpers um einen festen Punkt,” Math. Ann. 37 (2), 153–181 (1890).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, 83114, Donetsk, Ukraine

    G. V. Gorr

Authors
  1. G. V. Gorr
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. V. Gorr.

Additional information

Translated by A. Muravnik

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gorr, G.V. Analog of the Poinsot Interpretation of the Euler Solution in the Motion Problem for a Rigid Body in a Potential Field of Forces. Mech. Solids 55, 932–940 (2020). https://doi.org/10.3103/S0025654420070109

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0025654420070109

Keywords:

Search

Navigation