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IssuesArchive of Issues2007-6pp.883-896

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Yu. M. Urman, "Irreducible tensors and their applications in problems of dynamics of solids," Mech. Solids. 42 (6), 883-896 (2007)
Year 2007 Volume 42 Number 6 Pages 883-896
Title Irreducible tensors and their applications in problems of dynamics of solids
Author(s) Yu. M. Urman (Nizhnii Novgorod State Pedagogical University, Ulyanova 1, Nizhnii Novgorod, 603005, Russia, urman37@mail.ru)
Abstract One difficulty encountered in solving mechanical problems with complicated interaction is to express either the moments of forces or the force function via the phase variables of the problem. Here various transformations of coordinate systems are used, because interactions are determined by a relation between tensor variables one of which refers to the body and the other refers to the field. In this connection, the usual definition of a tensor in Cartesian coordinates is inconvenient because of the fact that the components of a tensor of rank l≥2 can be arranged as several linear combinations that behave differently under rotations of the coordinate system. Naturally, one needs to define tensors in such a way that their components and linear combinations of these be transformed in a unified manner under rotations of the coordinate system. This requirement is satisfied by irreducible tensors. The mathematical apparatus of irreducible tensors was created to satisfy the requirements of quantum mechanics and turned out to be rather universal. As far as the author knows, this apparatus was first used in mechanics by G. G. Denisov and the author of the present paper [1]. Using this apparatus, one can see the clear physical meaning of complicated interactions, express these interactions in invariant form, easily perform transformations from one coordinate system to another coordinate system turned relative to the first, consider rather complicated types of interactions writing them in compact form explicitly depending on the phase variables of the problem, easily use the symmetry of both the rigid body and the force field structure, and perform the averaging procedure for the entire object rather than componentwise. The present paper further develops the paper [1]. We present a brief introduction to the theory of irreducible tensors. We show that the force function of various interactions between a rigid body and a force field can be represented as the scalar product of irreducible tensors. We study general properties of evolution motions of a rigid body in axisymmetric and nonsymmetric force fields under the action of moments caused by various harmonics of the force function.
References
1.  G. G. Denisov and Yu. M. Urman, "Precessional Motions of Rigid Body Acted on by Moments That Have a Force Function," Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, No. 6, 5-14 (1975) [Mech. Solids (Engl. Transl.) 10 (6), 1-9 (1975)].
2.  D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (Nauka, Moscow-Leningrad, 1975; World Scientific, Singapore, 1988).
3.  Yu. M. Urman, "Invariant Representation of the Potential Interaction between a Rigid Body with Nonuniform Electromagnetic Field," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 1, 13-19 (2005) [Mech. Solids (Engl. Transl.) 40 (1), 8-13 (2005)].
4.  Yu. M. Urman, "The Barnett-London Effect and Its Influence on the Angular Motions of a Rigid Body in a Magnetic Field," Zh. Tekh. Fiz. 56 (11), 2081-2086 (1986).
5.  Yu. M. Urman, "Invariant Decomposition of the Force Function of the Mutual Attraction of a System of Bodies," Astron. Zh. 66 (5), 1081-1092 (1989).
6.  Yu. M. Urman, "Addition Theorems for Tensor Spherical Functions," Zh. Tekh. Fiz. 51 (3), 457-462 (1981).
7.  Yu. M. Urman, "Use of the Method of Irreducible Tensors in Problems of Celestial Mechanics," Astron. Zh. 72 (4), 596-603 (1995).
8.  Yu. M. Urman, "Method of Irreducible Tensors in Problems on Evolution Motions of a Rigid Body with a Fixed Point," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 4, 10-20 (1997) [Mech. Solids (Engl. Transl.) 32 (4), 7-15 (1997)].
9.  G. G. Denisov and V. N. Komarov, "Nonconservative Moments and Their Effect on Precession of a Noncontact Gyroscope," Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, No. 3, 15-23 (1979) [Mech. Solids (Engl. Transl.) 14 (3), 12-19 (1979)].
10.  Yu. G. Martynenko, Motion of a Rigid Body in Electric and Magnetic Fields (Nauka, Moscow, 1988) [in Russian].
11.  D. M. Klimov, G. N. Kosmodem'yanskaya, and F. L. Chernous'ko, "Motion of a Gyroscope with Contactless Suspension," Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, No. 2, 3-8 (1972) [Mech. Solids (Engl. Transl.)].
12.  F. L. Chernous'ko, "On the Motion of a Satellite about the Center of Mass under the Action of Gravitational Moments," Prikl. Mat. Mekh. 27 (3), 474-483 (1963) [J. Appl. Math. Mech. (Engl. Transl.) 27 (3), 708-722 (1963)].
13.  V. V. Beletskii, Artificial Satellite Motion about Its Center of Mass (Nauka, Moscow, 1965) [in Russian].
14.  L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 1, Mechanics (Nauka, Moscow, 1965; Pergamon Press, Oxford, 1969).
15.  V. V. Beletskii, "Dynamics of Fast Rotations," Nauchn. Tr. Inst. Mekh. Mosk. Gos. Univ., No. 29, 97-118 (1973).
16.  V. Volterra, "Sur la Theorie des Variations des Latitudes," Acta Math. 22, 201-358 (1899).
Received 20 January 2007
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