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Yu. M. Urman, "Invariant representation of the potential interaction between a rigid body with nonuniform electromagnetic field," Mech. Solids. 40 (1), 8-13 (2005)
Year 2005 Volume 40 Number 1 Pages 8-13
Title Invariant representation of the potential interaction between a rigid body with nonuniform electromagnetic field
Author(s) Yu. M. Urman (Nizhni Novgorod)
Abstract Problems of motion of rigid bodies in nonuniform force fields of various physical nature are characterized by a much more complex structure of body-field interaction as compared with the classical problem of motion of a rigid body with a fixed point in the uniform gravity field. In such problems, it is essential to take into account the torques described by tensor interaction. An even more complex interaction is observed in the case of contactless suspension with field sources surrounding the rotor from all sides close to its surface.

In view of this, it is desirable to have a model of the body-field interaction that could be conveniently used in investigations. Moreover, since the complex structure of the torques practically prevents an exact analytic investigation of all motions of a body in the force field, the said model should also be convenient for finding and studying approximate equations governing nonstationary motions of a body.

Finding the force function for the interaction of a body of an arbitrary shape with a field is usually a very difficult task, since one has to solve a boundary value problem of mathematical physics in a domain the boundary of which has a complex structure. In this paper, it is shown how, using the technique of irreducible tensors, one can immediately obtain an invariant representation of the force function for the body-field interaction without solving the boundary value problem.

This method is demonstrated for a body of an arbitrary shape with nonhomogeneous magnetic or electric properties interacting with a nonuniform electric or magnetic field. The general method is justified by examples for which specific calculations can be done. In the case of small dielectric or magnetic susceptibility of the rigid body, an analytical expression is obtained for the force function and the torques due to the interaction between various field harmonics and the body. In the case of an arbitrary dielectric susceptibility, the force function is obtained for a spherical body in a nonuniform field and an elliptic body in a uniform field.
References
1.  G. G. Denisov and Yu. M. Urman, "Precession motions of a rigid body subjected to torques that have a force function," Izv. AN SSSR. MTT [Mechanics of Solids], No. 6, pp. 5-14, 1975.
2.  Yu. M. Urman, "An invariant expansion of the force function for mutual attraction of a system of bodies," Astron. Zh., Vol. 66, No. 5, pp.  1081-1092, 1989.
3.  Yu. M. Urman, "Addition theorems for tensor spherical wave functions," Zh. Tekhn. Fiz., Vol. 51, No. 3, pp. 457-462, 1981.
4.  Yu. M. Urman, "Application of the method of irreducible tensors in problems of evolutionary motion of a rigid body with a fixed point," Izv. AN. MTT [Mechanics of Solids], No. 4, pp. 10-20, 1997.
5.  A. I. Vorob'ev and Yu. M. Urman, "Forces and perturbation torques acting on the rotor of a cryogenic gyroscope," Izv. AN SSSR. MTT [Mechanics of Solids], No. 2, pp. 15-23, 1984.
6.  D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of the Angular Momentum [in Russian], Nauka, Moscow, 1975.
7.  L. D. Landau and E. M. Lifshitz, Theoretical Physics. Volume 8. Electrodynamics of Continuous Media [in Russian], Nauka, Moscow, 1982.
8.  M. L. Levin and R. Z. Muratov, "Dielectric ellipsoid in a nonuniform static field," Zh. Tekhn. Fiz., Vol. 47, No. 12, pp. 2464-2471, 1977.
Received 26 November 2002
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