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.
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References
D. Lewis, T. Ratiu, J. C. Ratiu, and J. E. Marsden, “The heavy top: A geometric treatment,” Nonlinearity, Vol. 5, No. 1, 1–48 (1992).
J. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, Springer, New York (1999).
S. S. Zub, “Stable orbital motion of magnetic dipole in the field of permanent magnets,” Physica D: Nonlinear Phenomena, Vol. 275, 67–73 (2014).
H. R. Dullin, “Poisson integrator for symmetric rigid bodies,” Regular and Chaotic Dynamics, Vol. 9, No. 3, 255–264 (2004).
R. Abraham and J. Marsden, Foundations of Mechanics, American Mathematical Soc., Massachusetts (2002).
S. S. Zub, “Lie group as a configuration space for a simple mechanical system [in Russian],” Journal of Numerical and Applied Mathematics, Vol. 112, No. 2, 89–99 (2013).
S. S. Zub, “The canonical Poisson structure on T * SE (3) and Hamiltonian mechanics of the rigid body: Magnetic dipole dynamics in an external field,” Bulletin of NAS of Ukraine, No. 4, 25–31 (2013).
R. Abraham, J. E. Marsden, and T. S. Ratiu, Manifolds, Tensor Analysis, and Applications, Springer, New York (1988).
J. E. Marsden, G. Misiolek, J.-P. Ortega, M. Perlmutter, and T. S. Ratiu, Hamiltonian Reduction by Stages, Springer, New York (2007).
R. Zulanke and P. Vintgen, Differential Geometry and Vector Bundles [Russian translation], Mir, Moscow (1975).
J. M. Souriau, Structure of Dynamical Systems, Birkhauser, Boston (1997).
L. Grigoryeva, J.-P. Ortega, and S. S. Zub, “Stability of Hamiltonian relative equilibria in symmetric magnetically confined rigid bodies,” The Journal of Geometric Mechanics, Vol. 6, No. 3, 373–415 (2014).
S. I. Lyashko and V. V. Semenov, “Controllability of linear distributed systems in classes of generalized actions,” Cybernetics and Systems Analysis, Vol. 37, No. 1, 13–32 (2001).
S. I. Lyashko and D. A. Nomirovskii, “Generalized solutions and optimal controls in systems describing the dynamics of a viscous stratified fluid,” Differential Equations, Vol. 39, No. 1, 90–98 (2003).
S. I. Lyashko and D. A. Nomirovskii, “The generalized solvability and optimization of parabolic systems in domains with thin low-permeable inclusions,” Cybernetics and Systems Analysis, Vol. 39, No. 5, 737–745 (2003).
N. I. Lyashko, A. E. Grishchenko, and V. V. Onotskii, “A regularization algorithm for singular controls of parabolic systems” Cybernetics and Systems Analysis, Vol. 42, No. 1, 75–82 (2006).
D. A. Klyushin, N. I. Lyashko, and Yu. N. Onopchuk, “Mathematical modeling and optimization of intratumoral drug transport,” Cybernetics and Systems Analysis, Vol. 43, No. 6, 886–892 (2007).
A. A. Chikriy, G. Ts. Chikrii, and K. Yu. Volyanskyi, “Quasilinear positional integral games of approach,” Journal of Automation and Information Sciences, Vol. 33, No. 10, 31–52 (2001).
J.-P. Ortega and T. S. Ratiu, “Non-linear stability of singular relative periodic orbits in Hamiltonian systems with symmetry,” J. Geom. Phys., Vol. 32, No. 2, 160–188 (1999).
S. S. Zub, “Magnetic levitation in Orbitron system,” Problems of Atomic Science and Technology, Vol. 93, No. 5, 31–34 (2014).
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Translated from Kibernetika i Sistemnyi Analiz, No. 3, May–June, 2017, pp. 3–17.
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Zub, S.I., Zub, S.S., Lyashko, V.S. et al. Mathematical Model of Interaction of a Symmetric Top with an Axially Symmetric External Field. Cybern Syst Anal 53, 333–345 (2017). https://doi.org/10.1007/s10559-017-9933-7
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DOI: https://doi.org/10.1007/s10559-017-9933-7