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About the periodic solutions of a rigid body in a central Newtonian field. (English) Zbl 0773.70005

Summary: The equations of motion of a rigid body about a fixed point in a central Newtonian field is reduced to the equation of plane motion under the action of potential and gyroscopic forces, using the isothermal coordinates on the inertia ellipsoid. The construction of periodic solutions near the equilibrium points, by using the Lyapunov theorem for holomorphic integral, is obtained and the necessary and sufficient conditions for the stability of the system are given.

MSC:

70E15 Free motion of a rigid body
70E05 Motion of the gyroscope
70K20 Stability for nonlinear problems in mechanics
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References:

[1] Koloosoff, G.V.: 1903, ?On Some Modification of Hamilton’s Principle, Applied to the Problem of Mechanics of a Rigid Body?,Tr. Otd. Fizich. nauk Ob-va liubit. estestvozn 11, 5, 5-13.
[2] Vagner, E.A. and Demin, V.G.: 1975, ?On a Class of Periodic Motions of a Solid Body About a Fixed Point?,Prikl. Mat. Mekh. 39, No. 5. · Zbl 0349.70003
[3] Clebch, A.: 1970, ??ber die Bewegung eines Korpers in einer Flussing Flussigkeit?,A. Math. An. 2, 238-262.
[4] Kobb, M.G.: 1985, ?Sur la Probleme de la Rotation d’un Corps Autour d’un Point Fixe?,Bulletin de la Soci?t? Math. de France 23, 210-215. · JFM 26.0842.03
[5] Harlamova, E.I.: 1959, ?On the Equation of Motion for a Heavy Body with a Fixed Point?,Izv. Sibirsk, Otd ? Akasd Nauk SSSR 6, 7-17.
[6] Stekloff, V.A.: 1902, ?On Case of Motion of a Heavy Rigid Body with a Fixed Point?,Tr. Otd. Fizich. nauk ob-va liubit. estestvozn 8, 2, 526-528.
[7] Beletskii, V.V.: 1955, ?The Motion of an Artificial Satellite with Respect to its Center of Mass?, Nauk, Moscow.
[8] Arkangeleskii, Yu.A.: 1963, ?On the Algebraic and Single-Valued Integrals in the Problem of Motion of a Rigid Body in a Newtonian Force Field?,Prikl. Mat. Mek. 27, 697-698.
[9] Arkangeleskii, Yu.A.: 1963, ?On the Algebraic Integrals in the Problem of Motion of a Rigid Body in a Newtonian Field of Force?,Prikl. Mat. Mek. 27, 171-175.
[10] Demin, V.G. and Kiselev, F.I.: 1974, ?A New Class of Periodic Motion of a Rigid Body with One Fixed Point in a Newtonian Force Field?,Dokl. Akad. Nauk SSSR 214, 5, 270-272. · Zbl 0296.70005
[11] Liapunov, A.: 1966,Stability of Motion, Academic Press, New York.
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