Abstract
We consider a special case of the Euler–Poisson system describing the motion of a rigid body with a fixed point. It is the autonomous ODE system of sixth order with one parameter. Among the stationary points of the system we select two one-parameter families with resonance (0,0,λ,–λ,2λ,–2λ) of eigenvalues of the matrix of the linear part. For the stationary points, we compute the resonant normal form of the system using a program based on the MATHEMATICA package. Our results show that in cases of the existence of an additional first integral of the system its normal form is degenerate. So we assume that the integrability of a system can be checked through its normal form.
This is a preview of subscription content, log in via an institution to check access.
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
Bruno, A.D.: The normal form of differential equations. Doklady Akad. Nauk. SSSR 157, 1276–1279 (1964) (in Russian); Soviet Math. Doklady 5, 1105–1108 (1964) (in English)
Bruno, A.D.: Analytical form of differential equations. Trudy Mosk. Mat. Obshch. 25, 119–262 (1971); 26, 199–239 (1972)(in Russian); Trans. Moscow Math. Soc. 25, 131–288 (1971); 26, 199–239 (1972) (in English)
Bruno, A.D.: Local Methods in Nonlinear Differential Equations. Springer, Berlin (1989)
Bruno, A.D.: Normal forms. Mathematics and Computers in Simulation 45(5-6), 413–427 (1998)
Bruno, A.D.: Power Geometry in Algebraic and Differential Equations. Elsevier Science B. V., Amsterdam (2000)
Golubev, V.V.: Lectures on Integration of Equations of Motion of a Rigid Body Around a Fixed Point. Moscow, GITTL (1953) (in Russian)
Edneral, V.F.: A symbolic approximation of periodic solutions of the Henon–Heiles system by the normal form method. Mathematics and Computers in Simulation 45(5-6), 445–463 (1998)
Edneral, V.F., Khanin, R.: Application of the resonant normal form to high order nonlinear ODEs using MATHEMATICA. Nuclear Instruments and Methods in Physics Research A 502(2-3), 643–645 (2003)
Starzhinsky, V.M.: Applied Methods in Nonlinear Oscillation, ch. IX, Nauka, Moscow (1977) (in Russian)
Ziglin, S.L.: Branching solutions and nonexistence of integrals in the Hamiltonian mechanics I, II. Functional Analysis and its Applications 16(3), 30–41 (1982); 17(1), 8–23 (1983)
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
Bruno, A.D., Edneral, V.F. (2005). Normal Forms and Integrability of ODE Systems. 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_6
Download citation
DOI: https://doi.org/10.1007/11555964_6
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)