Abstract
We consider the non-canonical Hamiltonian dynamics of a gyrostat in Newtonian interaction with n spherical rigid bodies. Using the symmetries of the system we carry out two reductions. Then, working in the reduced problem, we obtain the equations of motion, a Casimir function of the system and the equations that determine the relative equilibria. Global conditions for existence of relative equilibria are given. Besides, we give the variational characterization of these equilibria and three invariant manifolds of the problem; being calculated the equations of motion in these manifolds, which are described by means of a canonical Hamiltonian system. We give some Eulerian and Lagrangian equilibria for the four body problem with a gyrostat. Finally, certain classical problems of Celestial Mechanics are generalized.
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References
Barkin, Y. V. and Demin, V. G.: 1982, Translatory Rotatory Motion of Celestial Bodies, Vol. 20, Itogi Nauki i Tekhniki, Seria Astronomia. VINITI, Moscow.
R. Cid A. Vigueras (1985) ArticleTitle‘About the problem of motion of N gyrostats: I. the first integrals’ Celest. Mech. & Dyn. Astron. 36 155–162 Occurrence Handle809857 Occurrence Handle1985CeMec..36..155C
F. Cosson B. Elmabsout (1997) ArticleTitle‘Relative equilibrium in the three-body problem with a rigid body’ Celest. Mech. & Dyn. Astron. 69 293–315 Occurrence Handle10.1023/A:1008225105537
R. Cushmann L. Bates (1987) Global Aspects of Classical Integrable Systems Birkhauser Basel
G.N. Duboshin (1984) ArticleTitle‘The problem of three rigid bodies’ Celest. Mech. & Dyn. Astron. 33 31–47 Occurrence Handle753166 Occurrence Handle1984CeMec..33...31D
E. Leimanis (1965) The General Problem of the Motion of Coupled Rigid Bodies About A Fixed Point Springer Verlag Berlin
A. Maciejewski (1995) ArticleTitle‘Reduction, relative equilibria and potential in the two rigid bodies problem’ Celest. Mech. & Dyn. Astron. 63 1–28 Occurrence Handle0883.70007 Occurrence Handle1386965 Occurrence Handle10.1007/BF00691912 Occurrence Handle1995CeMDA..63....1M
Marsden, J. E.: 1992, Lectures on Mechanics, L. M. S., Lectures Note Series 174, Cambridge University Press
J.E. Marsden T.S. Ratiu (1999) Introduction to Mechanics and symmetry Springer Verlag New York
J.E. Marsden T.S. Ratiu A. Weinstein (1984a) ArticleTitle‘Semidirect products and reductions in mechanics’ Trans. AMS 281 147–177 Occurrence Handle719663 Occurrence Handle10.2307/1999527
J.E. Marsden T.S. Ratiu A. Weinstein (1984b) ArticleTitle‘Reductions and Hamiltonian structures on duals of semidirect product Lie Algebras’ Cont. Math. 28 55–100 Occurrence Handle751975
J.E. Marsden A. Weinstein (1974) ArticleTitle‘Reductions of symplectic manifolds with symmetry’ Rep. Math. Phys. 5 121–130 Occurrence Handle402819 Occurrence Handle10.1016/0034-4877(74)90021-4
F. Mondéjar A. Vigueras (1999) ArticleTitle‘The Hamiltonian dynamics of the two gyrostats problem’ Celest. Mech. & Dyn. Astron. 73 303–312 Occurrence Handle10.1023/A:1008375820146 Occurrence Handle1999CeMDA..73..303M
F. Mondéjar A. Vigueras S. Ferrer (2001) ArticleTitle‘Symmetries, reduction and relative equilibria for a gyrostat in the three-body problem’ Celest. Mech.& Dyn. Astron. 81 45–50 Occurrence Handle10.1023/A:1013303002722 Occurrence Handle2001CeMDA..81...45M
J.A. Vera (2004) Reducciones, equilibrios y estabilidad en dinámica de sólidos rígidos y giróstatos Universidad Politécnica de Cartagena Spain
Vera, J. A. and Vigueras, A.: 2004, ‘Reduction, relative equilibria and stability for a gyrostat in the n-body problem’, In: M. C. López de Silanes et al. (eds), Monografías del Seminario Matemático García de Galdeano (VIII Journées Zaragoza-Pau de Mathematiques Appliquées et de Statistiques), Vol. 31, 257–271, Servicio de Publicaciones de la Universidad de Zaragoza, Zaragoza, Spain.
V.V. Vidiakin (1977) ArticleTitle‘Euler solutions in the problem of translational-rotational motion of three-rigid bodies’ Celest. Mech. & Dyn. Astron. 16 509–526 Occurrence Handle1977CeMec..16..509V
L.S. Wang P.S. Krishnaprasad J.H. Maddocks (1991) ArticleTitle‘Hamiltonian dynamics of a rigid body in a central gravitational field’ Celest. Mech. & Dyn. Astron. 50 349–386 Occurrence Handle1130692 Occurrence Handle10.1007/BF02426678 Occurrence Handle1991CeMDA..50..349W
L.S. Wang P.S. Krishnaprasad (1992) ArticleTitle‘Gyroscopic control and stabilization’ J. Nonlinear Sci. 2 367–415 Occurrence Handle1193421 Occurrence Handle10.1007/BF01209527
L.S. Wang K.Y. Lian P.T. Chen (1995) ArticleTitle‘Steady motions of gyrostat satellites and their stability’ IEEE Trans. Automat. Control 40 IssueID10 1732–1743 Occurrence Handle1354514 Occurrence Handle10.1109/9.467678
S.G. Zhuravlev A.A. Petrutskii (1990) ArticleTitle‘Current state of the problem of translational-rotational motion of three-rigid bodies’ Soviet Astron. 34 299–304 Occurrence Handle1990SvA....34..299Z
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Vera, J.A., Vigueras, A. Hamiltonian Dynamics of a Gyrostat in the N-Body Problem: Relative Equilibria. Celestial Mech Dyn Astr 94, 289–315 (2006). https://doi.org/10.1007/s10569-005-5910-y
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DOI: https://doi.org/10.1007/s10569-005-5910-y