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Equivalent problem in rigid body dynamics. I. (English) Zbl 0658.70008

In the present paper the problem of motion of a rigid body about a fixed point under the action of a stationary non-symmetric potential and gyroscopic forces is considered. The equations of motion in the vector Euler-Poisson form are derived. The relative motion of a satellite- gyrostat in its orbital system is a representative example. The interior forces due to the rotation of the orbital system are replaced by potential and Lorentz interactions between a combination of three classical fields and distributions of electric charges and magnetization attached to the mass of the body. The case when the body possesses an axis of symmetry passing through the fixed point is also considered. It is shown that this problem is equivalent to another one in which the body possesses spherical dynamical symmetry and the forces are symmetric around a space axis passing through the fixed point.
Having in mind the known integrable cases of the last problem, for the original one a pair of general integrable cases is constructed. One of these cases is a non-significant generalization of a classical result due to Brun and the other is new. A class of particular integrable cases depending on two arbitrary functions is also found. The introduced earlier transformation for the case of axially symmetric forces is used in the paper to generate classes of integrable cases depending on an arbitrary function from each of the mentioned cases. The equations of motion are also reduced to a single equation of the second order.
Reviewer: C.Mladenova

MSC:

70E15 Free motion of a rigid body
76E05 Parallel shear flows in hydrodynamic stability

Citations:

Zbl 0658.70009
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References:

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