Abstract
We consider Hamiltonian autonomous systems with n degrees of freedom near a singular point. In the case of absence of resonances of order less than or equal to 4 we present a direct computation of the Birkhoff normal form. In the case of two degrees of freedom, we study 1–parameter deformations of the 0 : 1, 1 : 1 and 2 : 1 resonant singularities. The obtained results are used in a direct derivation of the Deprits formula for the isoenergetic degeneracy determinant in the restricted three–body problem.
Article PDF
Similar content being viewed by others
References
Arnol’d V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations. Springer, New York (1983)
V. I. Arnol’d, Mathematical Methods of Classical Mechanics. 2nd ed. Graduate Texts in Mathematics 60, Springer, New York, 1989.
V. I. Arnol’d, V. V. Kozlov and A. I. Neishtat, Mathematical Aspects of Classical and Celestial Mechanics (Dynamical Systems III). Encyclopaedia Math. Sci. 3, 2006, Springer, New York, 1988 [Original Russian edition published by URSS, Moscow, 2002, 3rd ed., 2006].
Baider A., Sanders J.A.: Unique normal forms: The nilpotent Hamiltonian case. J. Differential Equations 92, 282–304 (1991)
Barwicz W., Żołaͅdek H.: The restricted three body problem revisited. J. Math. Anal. Appl. 366, 663–672 (2010)
Birkhoff G.D.: Dynamical Systems. Amer. Math. Soc. Providence, RI (1927)
H. Broer, I. Hoveijn, G. Lunter and G. Vegter, Bifurcations in Hamiltonian Systems. Computing Singularities by Gröbner Bases. Lecture Notes in Math. 1806, Springer, Berlin, 2003.
Deprit A., Deprit-Bartholomé A.: Stability of the triangular Lagrangian points. Astronomical J. 72, 173–179 (1967)
Duistermaat J.J.: The monodromy in the Hamiltonian Hopf bifurcation. J. Angew. Math. Phys. 49, 156–161 (1998)
E. A. Grebenikov, D. Kozak-Skoworodkina and M. Jakubiak, Computer Algebra Methods in the Many Body Problem. Izdat. Ross. Universiteta Druzhby Narodov, Moskva, 2001 (in Russian).
Leontovich A.M.: On stability of the Lagrangian periodic solutions of the restricted three body problem. Dokl. Akad. Nauk SSSR 143, 525–528 (1962) (in Russian)
Markeev A.P.: Libration Points in Celestial Mechanics and Cosmodynamics. Nauka, Moskva (1978) (in Russian)
Moser J.K.: Lectures on Hamiltonian systems. Mem. Amer. Math. Soc., 81, 60 (1968)
Prokopenya A.N.: Hamiltonian normalization in the restricted many-body problem by computer algebra methods. Program. Comput. Softw. 38, 156–166 (2012)
Siegel C.L., Moser J.K.: Lectures on Celestial Mechanics. Springer, New York (1971)
van der Meer J.-C.: Bifurcation at nonsemisimple 1 :–1 resonance. J. Angew. Math. Phys. 37, 425–437 (1986)
M. Wiliński, The restricted three body problem near the 1 : 1 resonance. Master Thesis, University of Warsaw, 2011 (Polish).
Author information
Authors and Affiliations
Corresponding author
Additional information
To Professors Bogdan Bojarski and Kazimierz Gęba
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
About this article
Cite this article
Barwicz, W., Wiliński, M. & Żołaͅdek, H. Birkhoff normalization, bifurcations of Hamiltonian systems and the Deprits formula. J. Fixed Point Theory Appl. 13, 587–610 (2013). https://doi.org/10.1007/s11784-013-0136-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11784-013-0136-1