Nonlinear Oscillations

, Volume 10, Issue 1, pp 62–77 | Cite as

Studying the stability of equilibrium solutions in a planar circular restricted four-body problem

  • E. A. Grebenikov
  • L. Gadomski
  • A. N. Prokopenya


The Newtonian circular restricted four-body problem is considered. We obtain nonlinear algebraic equations determining equilibrium solutions in the rotating frame and find six possible equilibrium configurations of the system. Studying the stability of equilibrium solutions, we prove that the radial equilibrium solutions are unstable, while the bisector equilibrium solutions are stable in Lyapunov’s sense if the mass parameter satisfies the conditions µ ∈ (0, µ0, where µ0 is a sufficiently small number, and µ ≠ µj, j = 1, 2, 3. We also prove that, for µ = µ1 and µ = µ3, the resonance conditions of the third order and the fourth order, respectively, are satisfied and, for these values of µ, the bisector equilibrium solutions are unstable and stable in Lyapunov’s sense, respectively. All symbolic and numerical calculations are done with the Mathematica computer algebra system.


Hamiltonian System Equilibrium Solution Canonical Transformation Celestial Mechanic Circular Restricted Problem 
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  1. 1.
    V. Szebehely, Theory of Orbits in the Restricted Problem of Three Bodies [Russian translation], Nauka, Moscow (1982).Google Scholar
  2. 2.
    A. P. Markeev, Libration Points in Celestial Mechanics and Astrodynamics [in Russian], Nauka, Moscow (1978).Google Scholar
  3. 3.
    A. P. Markeev, “Stability of Hamiltonian systems,” in: V. M. Matrosov, V. V. Rumyantsev, and A. V. Karapetyan (editors), Nonlinear Mechanics [in Russian], Fizmatlit, Moscow (2001), pp 114–130.Google Scholar
  4. 4.
    N. N. Bogolyubov and Yu. A. Mitropol’skii, Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Nauka, Moscow (1974).Google Scholar
  5. 5.
    Yu. A. Mitropol’skii, The Averaging Method in Nonlinear Mechanics [in Russian], Naukova Dumka, Kiev (1971).Google Scholar
  6. 6.
    L. M. Perko and E. L. Walter, “Regular polygon solutions of the N-body problem,” Proc. Amer. Math. Soc., 94, 301–309 (1985).MATHCrossRefGoogle Scholar
  7. 7.
    B. Elmabsout, “Sur l’existence de certaines configurations d’equilibre relatif dans le probleme des N corps,” Celest. Mech. Dynam. Astron., 41, 131–151 (1988).Google Scholar
  8. 8.
    E. A. Grebenikov, “New exact solutions in the plain symmetrical (n + 1)-body problem,” Rom. Astron. J., 7, 151–156 (1997).Google Scholar
  9. 9.
    E. A. Grebenikov, “Two new dynamical models in celestial mechanics,” Rom. Astron. J., 8, 13–19 (1998).Google Scholar
  10. 10.
    E. A. Grebenikov, D. Kozak-Skovorodkin, and M. Jakubiak, Methods of Computer Algebra in Many-Body Problem [in Russian], RUDN, Moscow (2002).Google Scholar
  11. 11.
    E. A. Grebenikov and A. N. Prokopenya, “Studying stability of the equilibrium solutions in the restricted Newton’s problem of four bodies,” Bul. Acad. Sti. Republ. Moldova. Mat., No. 2(42), 28–36 (2003).Google Scholar
  12. 12.
    A. N. Kolmogorov, “On the conservation of quasiperiodic motions for a small change in the Hamiltonian function,” Dokl. Akad. Nauk SSSR, 98, No. 4, 527–530 (1954).MATHGoogle Scholar
  13. 13.
    V. I. Arnold, “Small denominators and problems of stability of motion in classical and celestial mechanics,” Usp. Mat. Nauk, 18, No. 6, 91–192 (1963).Google Scholar
  14. 14.
    J. Moser, Lectures on Hamiltonian Systems [Russian translation], Mir, Moscow (1973).MATHGoogle Scholar
  15. 15.
    S. Wolfram, The Mathematica Book, Cambridge University Press, Cambridge (1996).MATHGoogle Scholar
  16. 16.
    A. M. Lyapunov, General Problem of Stability of Motion [in Russian], Gostekhizdat, Moscow (1950).Google Scholar
  17. 17.
    G. D. Birkhoff, Dynamical Systems [Russian translation], Gostekhteorizdat, Moscow (1941).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • E. A. Grebenikov
    • 1
  • L. Gadomski
    • 2
  • A. N. Prokopenya
    • 3
  1. 1.Computing CenterRussian Academy of SciencesMoscowRussia
  2. 2.University of PodlasieSiedlcePoland
  3. 3.Brest State Technical UniversityBrestBelarus

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