Abstract
A new solution is obtained for the problem concerning the motion of a rigid body having a fixed point, affected by potential and gyroscopic forces. With the use of a modified Poinsot method proposed by the author, it is shown that the body motion in the constructed solution can be presented by a nonslip rolling of the inertia ellipsoid of a body on a plane fixed in space. This result can be considered as an analog of a Poinsot result obtained based on the interpretation of motion a rigid body in Euler solution.
This is a preview of subscription content, log in via an institution to check access.
REFERENCES
L. Poinsot, “Thèorie nouvelle de la rotation des corps,” J. Math. Pures Appl. 1 (16), 289–336 (1851).
J. J. Sylvester, “On the motion of a rigid body acted on by no external forces,” Philos. Trans. R. Soc. London 156, 757–780 (1866).
J. Mac-Cullagh, “On the rotation of a solid body,” Proc. R. Irish Acad. 2, 542–545 (1840–1844); 3, 370–371 (1845–1847).
G. Darboux, “Sur la theorie de Poinsot et sur des mouvements correspondants a la meme polhodie,” C. R. Acad. Sci. 101, 1555–1561 (1885).
G. Darboux, “Sur le mouvement d’un corps pesant de revolution fixe par un point de son axe,” J. Math. Pures Appl. 1, 403–430 (1885).
C. G. J. Jacobi, “Sur la rotation d’un corps de rèvolution grave autour d’un point quelconque de son axe,” in Gesammelte Werke (G. Reimer, Berlin, 1882), Vol. 2, pp. 493–510.
W. Hess, “Über das Problem der Rotation,” Math. Ann. 20, 461–470 (1882).
W. Hess, “Über des Jacobische Theorem von der Ersetzbarkeit einer Lagrangeschen Rotation durch zwei Poinsotische Rotation,” Z. Math. Phys. 33, 292–305 (1888).
N. E. Zhukovskii, “A case of heavy solid motion near the fixed point: geometrical interpretation of a case observed by S. V. Kovalevskaya,” in Collection of Scientific Papers in 7 Vols. (Gostekhizdat, Leningrad, Moscow, 1948), Vol. 1, pp. 294–339 [in Russian].
G. K. Suslov, Theoretical Mechanics (Gostekhizdat, Moscow, 1946) [in Russian].
G. V. Gorr, L. V. Kudryashova, and L. A. Stepanova, Classical Problems on Rigid Body Dynamics (Naukova dumka, Kiev, 1978) [in Russian].
I. N. Gashenenko, G. V. Gorr, and A. M. Kovalev, Classical Problems on Rigid Body Dynamics (Naukova dumka, Kiev, 2012) [in Russian].
P. V. Kharlamov, “Kinematic interpretation of the motion of a body with a fixed point,” Prikl. Mat. Mekh. 28 (3), 502–507 (1964).
G. V. Gorr, “How to use Puansot theorem for kinematic interpreting a motion of a body with a fixed point,” Mekh. Tverd. Tela, No. 42, 26–36 (2012).
G. V. Gorr, “Analog of the Poinsot interpretation of the Euler solution in the motion problem for a rigid body in a potential feld of forces,” Mech. Solids 55 (7), 932–941 (2020).
G. V. Gorr, “On three invariant relations of the equations of motion of a body in a potential field of force,” Mech. Solids 54 (2), 234–244 (2020).
G. Grioli, “Questioni di dinamica del corpo rigido,” Atti. Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Natur. 35 (1-2), 35–39 (1963).
H. M. Yehia, “New generalizations of all the known integrable problems in rigid-body dynamics,” J. Phys. A: Math Gen. 32, 7565–7580 (1999).
H. M. Yehia, “Equivalent mechanical systems with cyclic coordinates and new integrable problems,” Int. J. Non-Linear Mech. 36, 89–105 (2001).
G. V. Gorr, “On a class of solutions for the dynamic equations of a rigid body acted upon by potential and gyroscopic forces,”Mech. Solids 57 (Suppl. 2), S107–S118 (2018).
T. Levi-Civita and U. Amaldi, Lezioni di Meccanica Razionale (Nicola Zanichelli Editore, Bologna, 1938), Vol. 2.
G. V. Gorr, Invariant Relations of Equations of Rigid Body Dynamics (Theory, Results, Comments) (Institute for Computer Science, Moscow, Izhevsk, 2017) [in Russian].
M. P. Kharlamov, “Symmetry in systems with gyroscopic forces,” Mekh. Tverd. Tela, No. 15, 87–93 (1983).
I. N. Gashenenko, “On Poinsot kinematic representation of a body motion in the Hess case,” Mekh. Tverd. Tela, No. 40, 12–20 (2010).
A. P. Markeev, “On Poinsot geometric interpretation of a rigid body motion in Euler case,” in Problems on Motion Control Mechanic for the Nonlinear Dynamical System. Intercollege Collection of Scientific Works (Perm, 1981), pp. 123–131 [in Russian].
G. V. Gorr, “An approach in studying gyrostat motion with variable gyrostatic moment,” Vestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki 31 (1), 1–14 (2021).
D. N. Goryachev, “New integrability cases for Euler dynamical equations,” Varshav. Univ. Izv., Book 3, 1–13 (1916).
D. N. Goryachev, “New cases of a rigid body motion round a fixed point,” Varshav. Univ. Izv., Book 3, 1–11 (1915).
H. M. Yehia, “New integrable problems in the dynamics of rigid bodies with Kovalevskaya configuration. I – the case of axisymmetric forces,” Mech. Res. Commun. 23 (5), 423–437 (1996).
A. V. Borisov and I. S. Mamaev, Rigid Body Dynamics (Regular and Chaotic Dynamics, Izhevsk, 2001) [in Russian].
I. V. Komarov and V. B. Kuznetsov, “Generalized Goryachev–Chaplygin gyrostate in quantum mechanics,” in Proc. Sci. Seminar of the Leningrad Branch of V. I. Steklov Mathematical Institute, USSR Acad. Sci. (Leningrad, 1987), Vol. 9, pp. 134–141.
I. V. Komarov and V. B. Kuznetsov, “Quasi-classical quantization of Kovalevskaya top,” Teor. Mat. Fiz. 73 (3), 335–347 (1987).
G. V. Gorr, “A complex approach to the interpretation of the motion of a solid with a fixed point,” Mech. Solids 56 (6), 932–946 (2021).
Funding
The study was supported by the Russian Science Foundation, project no. 19-71-30012.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The author of this work declares that he has no conflicts of interest.
Additional information
Translated by O. Polyakov
Publisher’s Note.
Allerton Press remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Gorr, G.V. A Kinematic Interpretation of Rigid Body Motion in a New Solution of Grioli Equations. Mech. Solids 58, 2738–2749 (2023). https://doi.org/10.3103/S0025654423080101
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0025654423080101