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Symbolic Computations of the Equilibrium Orientations of a System of Two Connected Bodies Moving on a Circular Orbit Around the Earth

Abstract

Computer algebra methods were used to find the roots of a nonlinear algebraic system that determines the equilibrium orientations for a system of two bodies, connected by a spherical hinge, that moves along a circular orbit under the action of gravitational torque. To determine the equilibrium orientations of two connected bodies the system of 12 algebraic equations was decomposed using algorithms for Gröbner basis construction. The number of equilibria was found by analyzing the real roots of the algebraic equations from the calculated Gröbner basis. Evolution of the conditions for equilibria existence in the dependence of the parameter of the problem was investigated. The effectiveness of the algorithms for Gröbner basis construction was analyzed depending on the number of parameters for the problem under consideration.

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Acknowledgements

The authors are grateful for the help that was provided in calculating the Gröbner basis on high performance computer by Professor Perepechko S.N. and we thank the reviewers for their detailed and useful comments, remarks and suggestions.

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Correspondence to Sergey A. Gutnik.

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Gutnik, S.A., Sarychev, V.A. Symbolic Computations of the Equilibrium Orientations of a System of Two Connected Bodies Moving on a Circular Orbit Around the Earth. Math.Comput.Sci. 15, 407–417 (2021). https://doi.org/10.1007/s11786-021-00511-6

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Keywords

  • Two connected bodies
  • Satellite–stabilizer system
  • Spherical hinge
  • Gravitational torque
  • Circular orbit
  • Lagrange equations
  • Algebraic equations
  • Equilibrium orientation
  • Computer algebra
  • Gröbner basis

Mathematics Subject Classification

  • 70-08