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

  • Dzmitry A. Budzko
  • Alexander N. Prokopenya
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7442)


We study stability of equilibrium positions in the spatial circular restricted four-body problem formulated on the basis of Lagrange’s triangular solution of the three-body problem. Using the computer algebra system Mathematica, we have constructed Birkhoff’s type canonical transformation, reducing the Hamiltonian function to the normal form up to the fourth order in perturbations. Applying Arnold’s and Markeev’s theorems, we have proved stability of three equilibrium positions for the majority of initial conditions in case of mass parameters of the system belonging to the domain of the solutions linear stability, except for the points in the parameter plane for which the third and fourth order resonance conditions are fulfilled.


Equilibrium Point Equilibrium Position Equilibrium Solution Canonical Variable Canonical Transformation 
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  1. 1.
    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
  2. 2.
    Szebehely, V.: Theory of Orbits. The Restricted Problem of Three Bodies. Academic Press, New York (1967)Google Scholar
  3. 3.
    Markeev, A.P.: Libration Points in Celestial Mechanics and Cosmodynamics. Nauka, Moscow (1978) (in Russian) Google 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)Google Scholar
  5. 5.
    Moser, J.: Lectures on the Hamiltonian Systems. Mir, Moscow (1973) (in Russian)Google Scholar
  6. 6.
    Wolfram, S.: The Mathematica Book, 4th edn. Wolfram Media/Cambridge University Press (1999)Google Scholar
  7. 7.
    Budzko, D.A., Prokopenya, A.N.: On the Stability of Equilibrium Positions in the Circular Restricted Four-Body Problem. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2011. LNCS, vol. 6885, pp. 88–100. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  8. 8.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Birkhoff, G.D.: Dynamical Systems. GITTL, Moscow (1941) (in Russian)Google Scholar
  10. 10.
    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
  11. 11.
    Liapunov, A.M.: General Problem about the Stability of Motion. Gostekhizdat, Moscow (1950) (in Russian)Google Scholar
  12. 12.
    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)MathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Dzmitry A. Budzko
    • 1
  • Alexander N. Prokopenya
    • 2
    • 3
  1. 1.Brest State UniversityBrestBelarus
  2. 2.Warsaw University of Live SciencesWarsawPoland
  3. 3.Collegium Mazovia Innovative University in SiedlceSiedlcePoland

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