Abstract
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.
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References
Szebehely, V.: Theory of Orbits. The Restricted Problem of Three Bodies. Academic Press, New York (1967)
Markeev, A.P.: Libration Points in Celestial Mechanics and Cosmodynamics. Nauka, Moscow (1978) (in Russian)
Whipple, A.L., Szebehely, V.: The restricted problem of n + ν bodies. Celestial Mechanics 32, 137–144 (1984)
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)
Moser, J.: Lectures on the Hamiltonian Systems. Mir, Moscow (1973) (in Russian)
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)
Wolfram, S.: The Mathematica Book, 4th edn. Wolfram Media/Cambridge University Press (1999)
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)
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)
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)
Liapunov, A.M.: General Problem about the Stability of Motion. Gostekhizdat, Moscow (1950) (in Russian)
Birkhoff, G.D.: Dynamical Systems. GITTL, Moscow (1941) (in Russian)
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)
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)
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Budzko, D.A., Prokopenya, A.N. (2011). 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) Computer Algebra in Scientific Computing. CASC 2011. Lecture Notes in Computer Science, vol 6885. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23568-9_8
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DOI: https://doi.org/10.1007/978-3-642-23568-9_8
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