Email this article Existance of Liouvillian solutions in the problem of motion of a heavy rigid body with a fixed point under the action of gyroscopic forces in the Hess case
- Authors: Kuleshov A.S.1, Skripkin A.D.1
-
Affiliations:
- Lomonosov Moscow State University
- Issue: Vol 89, No 3 (2025)
- Pages: 438-449
- Section: Articles
- URL: https://freezetech.ru/0032-8235/article/view/688570
- DOI: https://doi.org/10.31857/S0032823525030073
- EDN: https://elibrary.ru/JLKJAY
- ID: 688570
Cite item
Open Access
Access granted
Abstract
The paper studies the problem of motion of a rigid body about a fixed point under the action of gravity and gyroscopic forces in the Hess integrability case. It is shown, that the solution of the problem is reduced to the integration of the second – order linear differential equation with rational coefficients. Using the Kovacic algorithm, we obtain the conditions on the parameters of the problem under which we can find the general solution of the corresponding second order linear differential equation in explicit form. It is also shown that in the case when the rigid body with a fixed point moves under the action of only gyroscopic forces, the general solution of the corresponding linear differential equation can be found in explicit form for any values of parameters of the problem.
Full Text
About the authors
A. S. Kuleshov
Lomonosov Moscow State University
Author for correspondence.
Email: kuleshov@mech.math.msu.su
Russian Federation, Moscow
A. D. Skripkin
Lomonosov Moscow State University
Email: alexander.kuleshov@math.msu.ru
Russian Federation, Moscow
References
- Hess W. Ueber die Euler'schen Bewegungsgleichungen und ueber eine neue partikulare Losung des Problems der Bewegung eines starren Korpers um einen festen Punkt // Math. Ann., 1890, vol. 37, no. 2, pp. 153–181.
- Nekrasov P.A. On the problem of motion of a heavy rigid body about a fixed point // Mat. Sb., 1892, vol. 16, no. 3, pp. 508–517. (in Russian)
- Nekrasov P.A. Recherches analytiques sur un cas de rotation d'un solide pesant autour d'un point fixe // Math. Ann., 1896, vol. 47, pp. 445–530.
- Golubev V.V. Lectures on Integration of the Equations of Motion of a Rigid Body about a Fixed Point. Israel: Israeli Progr. for Sci. Transl., 1960. 287 p.
- Dokshevich A.I. Solutions in a Finite Form of the Euler – Poisson Equations. Kiev: Naukova Dumka, 1992. 168 p. (in Russian)
- Borisov A.V., Mamaev I.S. Rigid Body Dynamics. Berlin; Boston: Higher Education Press and Walter de Gruyter GmbH, 2019. 521 p.
- Gashenenko I.N., Gorr G.V., Kovalev A.M. Classical Problems in the Dynamics of Rigid Body. Kiev: Naukova Dumka, 2012. 401 p. (in Russian)
- Kovalev A.M. The moving angular velocity hodograph in Hess’ solution of the problem of motion of a body with a fixed point // JAMM, 1968, vol. 32, iss. 6, pp. 1129–1137.
- Kovalev A.M. Motion of a body in the Hess case // Mech. of Solids. Resp. Mezhved. Sb., 1969, vol. 1, pp. 12–27. (in Russian)
- Emel’yanova I.S. One case of solving the Hess problem in trigonometric functions // Rus. Math., 1998, vol. 42, no. 3, pp. 7–12.
- Borisov A.V., Mamaev I.S. The Hess case in rigid-body dynamics // JAMM, 2003, vol. 67, no. 2, pp. 227–235.
- Belyaev A.V. On the general solution of the problem of the motion of a heavy rigid body in the Hess case // Sb. Math., 2015, vol. 206, no. 5, pp. 621–649.
- Bizyaev I.A., Borisov A.V., Mamaev I.S. The Hess–Appelrot system and its nonholonomic analogs // Proc. Steklov Inst. Math., 2016, vol. 294, pp. 252–275.
- Gashenenko I.N. The periodic motions of a rigid body in the Hess case // Mech. of Solids. Resp. Mezhved. Sb., 2012, vol. 42, pp. 14–25.
- Kholostova O.V. On the dynamics of a rigid body in the Hess case at high-frequency vibrations of a suspension point // Rus. J. of Nonlin. Dyn., 2020, vol. 16, no. 1, pp. 59–84.
- Sretenskii L.N. Some integrability cases for the equations of gyrostat motion // Dokl. Akad. Nauk SSSR, 1963, vol. 149, no. 2, pp. 292–294.
- Lunev V.V. Integrable cases in the problem of the motion of a heavy rigid body with a fixed point in a Lorentz force field // Sov. Phys., Dokl., 1984, vol. 29, pp. 297–298.
- Samsonov V.A. On the rotation of a body in a magnetic field // Akademiia Nauk SSSR, Izvestiia, Mekhanika Tverdogo Tela. 1984, no. 4, pp. 32–34.
- Kozlov V.V. The problem of the rotation of a rigid body in a magnetic field // Akademiia Nauk SSSR, Izvestiia, Mekhanika Tverdogo Tela, 1985, no. 6, pp. 28–33.
- Kosov A.A. On analogues of the Hess case for a gyrostat under the action of the moment of gyroscopic and circular forces // Mech. Solids, 2022, vol. 57, no. 6, pp. 1848–1861.
- Kovacic J. An algorithm for solving second order linear homogeneous differential equations // J. Symb. Comp., 1986, vol. 2. pp. 3–43.
- Bardin B.S., Kuleshov A.S. The Kovacic Algorithm and its Applications to the Problems of Classical Mechanics. Moscow: Moscow Aviation Inst., 2020, 260 p. (in Russian)
- Kuleshov A.S. Application of the Kovacic algorithm to the study of the motion of a heavy rigid body with a fixed point in the Hess case // Itogi Nauki I Tekhn. Ser. Modern Mathem.&Its Appl. Thematic Rev., 2021, vol. 202, pp. 10–42.
- Bardin B.S., Kuleshov A.S. Application of the Kovacic algorithm for the investigation of motion of a heavy rigid body with a fixed point in the Hess case // ZAMM, 2022, vol. 102, no. 11.
Supplementary files
