On the Stability of Equilibrium Positions in the Circular Restricted Four-Body Problem

  • Dzmitry A. Budzko
  • Alexander N. Prokopenya
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6885)


We consider the stability of equilibrium positions in the planar circular restricted four-body problem formulated on the basis of Lagrange’s triangular solution of the three-body problem. The stability problem is solved in a strict nonlinear formulation on the basis of Arnold–Moser and Markeev theorems. Peculiar properties of the Hamiltonian normalization are discussed, and the influence of the third and fourth order resonances on stability of the equilibrium positions has been analyzed.


Equilibrium Point Equilibrium Position Equilibrium Solution Canonical Variable Canonical Transformation 
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  1. 1.
    Szebehely, V.: Theory of Orbits. The Restricted Problem of Three Bodies. Academic Press, New York (1967)Google Scholar
  2. 2.
    Markeev, A.P.: Libration Points in Celestial Mechanics and Cosmodynamics. Nauka, Moscow (1978) (in Russian)Google Scholar
  3. 3.
    Whipple, A.L., Szebehely, V.: The restricted problem of n + ν bodies. Celestial Mechanics 32, 137–144 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Arnold, V.I.: Small denominators and problems of stability of motion in classical and celestial mechanics. Uspekhi Math. Nauk. 18(6), 91–192 (1963) (in Russian)MathSciNetGoogle Scholar
  5. 5.
    Moser, J.: Lectures on the Hamiltonian Systems. Mir, Moscow (1973) (in Russian) Google Scholar
  6. 6.
    Markeev, A.P.: Stability of the Hamiltonian systems. In: Matrosov, V.M., Rumyantsev, V.V., Karapetyan, A.V. (eds.) Nonlinear Mechanics, pp. 114–130. Fizmatlit, Moscow (2001) (in Russian)Google Scholar
  7. 7.
    Wolfram, S.: The Mathematica Book, 4th edn. Wolfram Media/Cambridge University Press (1999)Google Scholar
  8. 8.
    Budzko, D.A.: Linear stability analysis of equilibrium solutions of restricted planar four-body problem. In: Gadomski, L., et al. (eds.) Computer Algebra Systems in Teaching and Research. Evolution, Control and Stability of Dynamical Systems, pp. 28–36. The College of Finance and Management, Siedlce (2009)Google Scholar
  9. 9.
    Budzko, D.A., Prokopenya, A.N.: Stability analysis of equilibrium solutions in the planar circular restricted four-body problem. In: Gadomski, L., et al. (eds.) Computer Algebra Systems in Teaching and Research. Differential Equations, Dynamical Systems and Celestial Mechanics, pp. 141–159. Wydawnictwo Collegium Mazovia, Siedlce (2011)Google Scholar
  10. 10.
    Budzko, D.A., Prokopenya, A.N.: Symbolic-numerical analysis of equilibrium solutions in a restricted four-body problem. Programming and Computer Software 36(2), 68–74 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Liapunov, A.M.: General Problem about the Stability of Motion. Gostekhizdat, Moscow (1950) (in Russian)Google Scholar
  12. 12.
    Birkhoff, G.D.: Dynamical Systems. GITTL, Moscow (1941) (in Russian)Google Scholar
  13. 13.
    Budzko, D.A., Prokopenya, A.N., Weil, J.A.: Quadratic normalization of the Hamiltonian in restricted four-body problem. Vestnik BrSTU. Physics, Mathematics, Informatics (5), 82–85 (2009) (in Russian)Google Scholar
  14. 14.
    Gadomski, L., Grebenikov, E.A., Prokopenya, A.N.: Studying the stability of equilibrium solutions in the planar circular restricted four-body problem. Nonlinear Oscillations 10(1), 66–82 (2007)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Dzmitry A. Budzko
    • 1
  • Alexander N. Prokopenya
    • 2
    • 3
  1. 1.Brest State UniversityBrestBelarus
  2. 2.Collegium Mazovia in SiedlceSiedlcePoland
  3. 3.Brest State Technical UniversityBrestBelarus

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