 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
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| A. P. Markeev, "Motion of a rigid body with a single fixed point in the Steklov case," Mech. Solids. 40 (1), 14-25 (2005) |
| Year |
2005 |
Volume |
40 |
Number |
1 |
Pages |
14-25 |
| Title |
Motion of a rigid body with a single fixed point in the Steklov case |
| Author(s) |
A. P. Markeev (Moscow) |
| Abstract |
A motion of a heavy rigid body with a single fixed point is analyzed. The
center of gravity of the body is located on the medium or the smallest
axis of the ellipsoid of inertia. In addition to the "triangle"
inequalities usual for a rigid body, the moments of inertia satisfy the
conditions B>A>2C or 2B>A>B>C and A>2C. Under these conditions, the
Euler-Poisson equations may have particular periodic solutions indicated
by V. A. Steklov. In this paper, we study the problem of orbital stability
of periodic motions of a rigid body which correspond to the Steklov
solutions. |
| References |
| 1. | V. A. Steklov,"A new particular solution of differential
equations of motion of a heavy rigid body with a fixed point," Trudy
Otdel. Fiz. Nauk Obsch. Lyubit. Estestvozn., Vol. 10, No. 1, pp. 1-3,
1899. |
| 2. | G. V. Gorr and L. V. Kudryashova, Classical Problems of Dynamics of
a Rigid Body: Development and State of the Art [in Russian], Naukova Dumka,
Kiev, 1978. |
| 3. | A. I. Dokshevich, Closed-form Solutions of Euler-Poisson Equations
[in Russian], Naukova Dumka, Kiev, 1992. |
| 4. | P. A. Kuz'min, "An addition to the V. A. Steklov case of motion of
a rigid body about a fixed point," PMM [Applied Mathematics and Mechanics], Vol. 16, No. 2, pp. 243-245,
1952. |
| 5. | P. A. Kuz'min, "Particular kinds of motion of a heavy rigid body
about a fixed point (in works by Russian scientists)," Trudy Kazan.
Aviats. In-ta, Vol. 27, pp. 91-121, 1953. |
| 6. | A. M. Zhuravskii, A Handbook on Elliptic Functions [in Russian],
Izd-vo AN SSSR, Moscow, Leningrad, 1941. |
| 7. | E. I. Kharlamova and G. V. Mozalevskaya, "Investigation of the
V. A. Steklov solution of the equations of motion of a body with a fixed
point," in Matematicheskaya Fizika [in Russian], No. 5, pp. 194-202,
Nauk. Dumka, Kiev, 1968. |
| 8. | A. A. Bogoyavlenskii, "On some particular cases of the motion of a
heavy rigid body with a fixed point," PMM [Applied Mathematics and Mechanics], Vol. 22, No. 5, pp. 622-645,
1958. |
| 9. | A. A. Bogoyavlenskii, "On some particular solutions of the problem
of motion of a heavy rigid body with a fixed point," PMM [Applied Mathematics and Mechanics], Vol. 22, No. 6,
pp. 738-749, 1958. |
| 10. | D. N. Goryachev, "A new particular solution of the problem of
motion of a heavy rigid body with a fixed point," Trudy Otdel. Fiz. Nauk
Obsch. Lyubit. Estestovzn., Vol. 10, No. 1, pp. 23-24, 1899. |
| 11. | S. A. Chaplygin, "A new particular solution of the problem of
rotation of a heavy rigid body about a fixed point," Trudy Otdel. Fiz.
Nauk Obsch. Lyubit. Estestovzn., Vol. 12, No. 1, pp. 1-4, 1904. |
| 12. | N. Kowalewski, "Eine neue particuläre Lösung der
Differentialgleichungen der Bewegung eines schweren starren Körpers um
einen festen Punkt," Math. Annalen., Bd. 65, S. 528-537, 1908. |
| 13. | P. V. Kharlamov, "Polynomial solutions of the equations of motion
of a body with a fixed point," PMM [Applied Mathematics and Mechanics], Vol. 29, No. 1, pp. 26-34, 1965. |
| 14. | L. V. Kudryashova and G. V. Mozalevskaya, "On a class of
solutions of the problem of motion of a body with a fixed point," in
Matematicheskaya Fizika [in Russian], No. 5, pp. 139-150, Naukova Dumka,
Kiev, 1968. |
| 15. | A. P. Markeev, "Algorithm of normalization of the Hamiltonian
system for the problem of the orbital stability of periodic motions,"
PMM [Applied Mathematics and Mechanics], Vol. 66, No. 6, pp. 929-9938, 2002. |
| 16. | V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical
Aspects of Classical and Celestial Mechanics [in Russian], Editorial URSS,
Moscow, 2002. |
| 17. | A. P. Markeev, Libration Points in Celestial Mechanics and
Cosmodynamics [in Russian], Nauka, Moscow, 1978. |
| 18. | A. P. Markeev, Theoretical Mechanics [in Russian], NITs "Regul.
Khaot. Din.," Izhevsk, 2001. |
| 19. | I. G. Malkin, Theory of Stability of Motion [in Russian], Nauka,
Moscow, 1966. |
| 20. | E. Yu. Kucher, "Characteristic parameters of Steklov and
Chaplygin periodic solutions," in Mekhanika Tverdogo Tela [in Russian],
No. 33, pp. 33-39, In-t Prikl. Metematiki i Mekhaniki NAN Ukrainy,
Donetsk, 2003. |
| 21. | A. P. Markeev, "One approach to the stability abalysis of
equilibrium states of Hamiltonian systems," Izv. AN. MTT [Mechanics of
Solids], No. 6, pp. 3-12, 2004. |
| 22. | A. P. Markeev, "On area preserving mappings and their
applications to the dynamics of systems with collisions," Izv. AN. MTT
[Mechanics of Solids], No. 2, pp. 37-54, 1996. |
|
| Received |
26 April 2004 |
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