Mathematical Model of Interaction of a Symmetric Top with an Axially Symmetric External Field
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CYBERNETICS MATHEMATICAL MODEL OF INTERACTION OF A SYMMETRIC TOP WITH AN AXIALLY SYMMETRIC EXTERNAL FIELD S. I. Zub,1 S. S. Zub,2† V. S. Lyashko,3 N. I. Lyashko,4 and S. I. Lyashko2‡
UDC 519.6:531:537
Abstract. A symmetric top is considered, which is a particular case of a mechanical top that is usually described by the canonical Poisson structure on T * SE (3). This structure is invariant under the right action of the rotation group SO(3) , but the Hamiltonian of the symmetric top is invariant only under the right action of the subgroup S 1 , which corresponds to the rotation of the symmetric top around its axis of symmetry. This Poisson structure is obtained as the reduction T * SE (3) / S 1 . A Hamiltonian and motion equations are proposed that describe a wide class of interaction models of the symmetric top with an axially symmetric external field. Keywords: mathematical model of a symmetric top, Poisson reduction, symplectic leaf, Kirillov–Kostant–Souriau 2-form, relative equilibrium, energy-momentum method. INTRODUCTION The mathematical model of the Lagrangian top is well-known long ago, but it is still a subject of inquiry [1]. The wide application of group-theoretical methods of Hamiltonian mechanics has determined a new stage of studying this classical model [1, 2]. The efficiency of the latter has manifested itself in investigating the stability of dynamic magnetic systems [3, 4]. In what follows, it is shown that the Hamiltonian reduction from the general asymmetric body to the symmetric top leads to the Lie–Poisson structure embedded in se(3) * . Symplectic leaves of this structure are orbits of the coadjoint representation of the group SE (3) . 1. HAMILTONIAN FORMALISM ON T * SO( 3) Representation of the right trivialization T * SO( 3) . Consider now some relations many of which are considered in [2, 5] and are useful for the group SO(3) and its cotangent bundle. The group SO(3) is formed by orthogonal unimodular matrices R , i.e., R T = R -1 , det ( R ) = 1. Accordingly, the Lie algebra so(3) is formed by antisymmetric 3 ´ 3 matrices with a Lie bracket in the form of a matrix commutator. 1
National Scientific Center “Institute of Metrology,” Ministry of Economic Development and Trade of Ukraine, Kharkiv, Ukraine, [email protected]. 2Taras Shevchenko National University of Kyiv, Kyiv, Ukraine, † [email protected]; ‡[email protected]. 3P. L. Shupyk National Medical Academy of Postgraduate Education, Kyiv, Ukraine. 4V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 3, May–June, 2017, pp. 3–17. Original article submitted July 5, 2016. 1060-0396/17/5303-0333 ©2017 Springer Science+Business Media New York
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^ We introduce an isomorphism of vector spaces : R 3 ® so(3) [2, p. 285] such that $Y kl = -e ikl x i and 1 x i = - e ikl $Y kl , where e ikl is the Levi–Chivita symbol. Then 2 ì $YR = Y ´ R; [ $Y, $R ] = $Y $R - $R $Y = Y ´ R; ï í 1 $ $ -1 ï á Y, Rñ = - tr ( Y $R );
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