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Symbolic-numerical methods for searching equilibrium states in a restricted four-body problem

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Abstract

Algorithms for searching equilibrium solutions of the circular restricted four-body problem formulated on the basis of the triangular Lagrange solutions of the three-body problem are discussed. An algorithm is proposed for calculating the bifurcation curve in the plane of system parameters that separates domains of the eight and ten equilibrium solutions. For the parameter values corresponding to the bifurcation curve, the system has nine equilibrium solutions. In the neighborhood of the bifurcation points, the equilibrium solutions are found in the form of power series in terms of a small parameter. The dependence of these solutions on the system parameters is studied numerically. Codes of algorithms implemented in the computer algebra system Mathematica are presented.

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References

  1. Duboshin, G.N., Nebesnaya mekhanika. Analiticheskie i kachestvennye metody (Celestial Mechanics: Analytical and Qualitative Methods), Moscow: Nauka, 1978.

    Google Scholar 

  2. Whipple, A.L. and Szebehely, V., The Restricted Problem of n + ν Bodies, Celestial Mechanics, 1984, vol. 32, pp. 137–144.

    Article  MathSciNet  MATH  Google Scholar 

  3. Szebehely, V., Theory of Orbits: The Restricted Problem of Three Bodies, New York: Academic, 1967.

    Google Scholar 

  4. Budzko, D.A. and Prokopenya, A.N., On the Stability of Equilibrium Positions in the Circular Restricted Four-Body Problem, Computer Algebra in Scientific Computing, Gerdt, V.P, Koepf, W., Mayr, E.W., and Vorozhtsov, E.V., Eds., LNCS, vol. 6885, Heidelberg: Springer, 2011, pp. 88–100.

    Chapter  Google Scholar 

  5. Wolfram, S., The Mathematica Book, Wolfram Media, Cambridge Univ. Press, 1999, 4th edition.

    MATH  Google Scholar 

  6. Budzko, D.A. and Prokopenya, A.N., Symbolic-Numerical Analysis of Equilibrium Solutions in a Restricted Four-Body Problem, Programming Comput. Software, 2010, vol. 36, no. 2, pp. 68–74.

    Article  MATH  Google Scholar 

  7. Simo, C., Relative Equilibrium Solutions in the Four Body Problem, Celestial Mechanics, 1978, vol. 18, pp. 165–184.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Brest State University, bul. Kosmonavtov 21, Brest, 224016, Belarus

    D. A. Budzko

  2. Warsaw University of Life Sciences, ul. Novoursynovska 166, Warsaw, 02-787, Poland

    A. N. Prokopenya

  3. Brest State Technical University, ul. Moskovskaya 267, Brest, 224017, Belarus

    A. N. Prokopenya

Authors
  1. D. A. Budzko
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  2. A. N. Prokopenya
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Correspondence to D. A. Budzko.

Additional information

Original Russian Text © D.A. Budzko, A.N. Prokopenya, 2013, published in Programmirovanie, 2013, Vol. 39, No. 2.

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Budzko, D.A., Prokopenya, A.N. Symbolic-numerical methods for searching equilibrium states in a restricted four-body problem. Program Comput Soft 39, 74–80 (2013). https://doi.org/10.1134/S0361768813020035

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  • DOI: https://doi.org/10.1134/S0361768813020035

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