Abstract
Using the computer algebra methods, different authors have proved the existence of new classes of the homografic solutions, in the Lagrange-Wintner sense [1], in the Newtonian many-body problem [2-6]. E.A. Grebenicov has also shown that any homographic solution of the n-body problem generates a new dynamical model, namely, “the restricted Newtonian (n + 1)-body problem”. These problems are similar to the famous “restricted three-body problem” which was proposed for the first time by K. Jacobi [7]. Then the theorems of the existence of stationary solutions (the equilibrium positions) for some fixed values of the parameter n were proved [8-10], and the problem of studying the stability of these solutions in the Lyapunov sense was formulated. The study of this problem can be done only on the basis of the KAM-theory [11-13] and only for the dynamical systems with two degrees of freedom it can be realized. We have shown that all the planar restricted n-body problems belong to this class for any n. The situation is essentially complicated for the critical (resonance) cases, when the eigenvalues of the linearized system of differential equations in the neighborhood of the stationary solution are rationally commensurable. In these critical cases the stability problem for hamiltonian dynamics may be studied only on the basis of Markeev and Sokolsky theorems [14,15]. These theorems contain mathematical estimations of the influence of so-called “non-annihilable resonance terms” in the Poincaré-Birkho. normalizing transformations, which must be taken into account in the theorems on the stability of stationary solutions of hamiltonian equations in critical cases [16].
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Wintner, A.: The Analytical Foundations of Celestial Mechanics. Princeton Univ. Press, Princeton (1941)
Elmabsout, B.: Sur l’existence de certaines configurations d’équillibre rélatif dans le problème des n corps. Celestial Mech. and Dynamical Astron. 4(1), 131–151 (1988)
Grebenicov, E.A.: The existence of the exact symmetrical solutions in the planar Newtonian many-body problem (in Russian). Mathematical Modeling 10(8), 74–80 (1998)
Zemtsova, N.I.: Stability of the stationary solutions of the differential equations of restricted Newtonian problem with incomplete symmetry. Nonlinear Dynamics and Systems Theory, Kiev 3(1), 105–116 (2003)
Bang, D., Elmabsout, B.: Configurations polygonales en equilibre relative. C.R. Acad. Sci., Série Iib. 329, 243–248 (2001)
Grebenicov, E.A., Prokopenya, A.N.: On the existence of a new class of the exact solutions in the planar Newtonian many-body problem (in Russian). In: Bereznev, V.A. (ed.) The Questions of Modeling and Analysis in the Problems of Making Decision, pp. 39–57. Computing Center RAS, Moscow (2004)
Jacobi, K.: Gesammelte Werke, Bd. 1-7, Berlin (1881–1891)
Grebenicov, E.A., Kozak-Skoworodkin, D., Jakubiak, M.: Methodes of Computer Algebra in Many-Body Problem, Ed. of UFP, Moscow (2002) (in Russian)
Siluszyk, A.: Problem on linear stability of stationary solutions of restricted 8-body problem with incomplete symmetry (in Russian). BRDU, Brest 2, 20–26 (2004)
Ikhsanov, E.V.: Stabilty of equilibrium state in restricted 10-body problem for resonance case of 4th order. In: Bereznev, V.A. (ed.) The Questions of Modeling and Analysis in the Problems of Making Decision (in Russian), pp. 16–23. Computing Center RAS, Moscow (2004)
Kolmogorov, A.N.: General theory of dynamical systems and classical mechanics (in Russian). In: Int. Math. Congress in Amsterdam, pp. 187–208. Physmatgiz, Moscow (1961)
Arnold, V.I.: About stability of equilibrium positions of Hamiltonian systems in general eliptic case (in Russian). DAN USSR 137, 255–257 (1961)
Moser, J.K.: Lectures on Hamiltonian Systems, p. 295. Courant Institute of Mathematical Science, New York (1968)
Markeev, A.P.: Libration Points in Celestial Mechanics and Cosmodynamics (in Russian), Nauka, Moscow (1974)
Sokol’sky, A.G.: About stability of Hamiltonian autonomy system with the resonance of first order (in Russian). J. Appl. Math. and Mech. 41, 24–33 (1977)
Birkhoff, G.D.: Dynamical Systems (in Russian), GITTL, Moscow (1941)
Wolfram, S.: The Mathematica – Book. Cambridge University Press, Cambridge (1996)
Lyapunov, A.M.: General Problem on Stability of Motion (in Russian), vol. 1. Academy of Sciences of the USSR, Moscow (1954)
Kozak-Skoworodkin, D.: The Stability of Equilibrium Points In Case of Resonance σ 1 = 2σ 2 in the Restricted Seven-Body Problem (in Russian). BRDU, Brest 1(38), 15–22 (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Grebenicov, E.A., Kozak-Skoworodkin, D., Jakubiak, M. (2005). Investigation of the Stability Problem for the Critical Cases of the Newtonian Many-Body Problem. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2005. Lecture Notes in Computer Science, vol 3718. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11555964_20
Download citation
DOI: https://doi.org/10.1007/11555964_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28966-1
Online ISBN: 978-3-540-32070-8
eBook Packages: Computer ScienceComputer Science (R0)