Abstract
In the paper [1], the special case of the Euler–Poisson equations describing movements of a heavy rigid body with a fixed point is considered. Among stationary points of the system, two of one-parameter families were chosen. These families correspond to the resonance of eigenvalues (0, 0, λ, − λ, 2λ, − 2λ) of the matrix of the linear part of the system, also in [1] it was conjectured the absence of the additional first integral (with respect to well-known 3 integrals (2)) near these families, except of classical cases of global integrability. In this paper, the supposition is proved by calculations of coefficients of the normal form.
The authors are supported by the Russian Foundation for Basic Research Grant No. 11-01-00023-a.
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References
Bruno, A.D.: Theory of Normal Forms of the Euler–Poisson Equations. Preprint No. 100, Keldysh Institute for Applied Mathematics of RAS, Moscow (2005), http://library.keldysh.ru/preprint.asp?lg=e&id=2005-100 (abstract), http://dl.dropbox.com/u/59058738/Preprint100.pdf (full text in Russian)
Golubev, V.V.: Lectures on the Integration of the Equation of Motion of a Rigid Body about of Fixed Point. Gostehizdat, Moscow (1953) (in Russian); NSF Israel Program for Scientific Translations, Washington (1960) (in English)
Bruno, A.D.: Power Geometry in Algebraic and Differential Equations. Nauka, Moscow (1998); Elsevier, Amsterdam (2000)
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)
Edneral, V.F.: An Algorithm for Construction of Normal Forms. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2007. LNCS, vol. 4770, pp. 134–142. Springer, Heidelberg (2007)
Bruno, A.D., Sadov, S.Y.: Formal integral of a divergentless system. Matem. Zametki 57(6), 803–813 (1995); Mathematical Notes 57(6), 565–572 (1995)
Bruno, A.D., Edneral, V.F.: Normal Forms and Integrability of ODE Systems. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2005. LNCS, vol. 3718, pp. 65–74. Springer, Heidelberg (2005)
Bruno, A.D., Edneral, V.F.: Normal form and integrability of ODE systems. Programmirovanie 32(3), 22–29 (2006); Programming and Computer Software 32(3), 1–6 (2006)
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Bruno, A.D., Edneral, V.F. (2012). Calculation of Normal Forms of the Euler–Poisson Equations. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2012. Lecture Notes in Computer Science, vol 7442. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32973-9_6
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DOI: https://doi.org/10.1007/978-3-642-32973-9_6
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