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Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2017, Volume 27, Issue 4, Pages 532–539 (Mi vuu606)  

MATHEMATICS

The Adler–van Moerbeke integrable case. Visualization of bifurcations of Liouville tori

S. V. Sokolov

Institute of Machines Science named after A.A. Blagonravov of the Russian Academy of Sciences, Malyi Khariton'evskii per., 4, Moscow, 101990, Russia

Abstract: In this paper we consider an integrable Hamiltonian system on the Lie algebra $so(4)$ with an additional integral of the fourth degree — the Adler–van Moerbeke integrable case. We discuss classical works which explore, on the one hand, the dynamics of a rigid body with cavities completely filled with an ideal fluid performing a homogeneous vortex motion and, on the other hand, are devoted to the study of geodesic flows of left-invariant metrics on Lie groups. The equations of motion, the Hamiltonian function, Lie–Poisson brackets, Casimir functions and the phase space of the case under consideration are given. In previous papers, the investigation of the phase topology of the integrable Adler-van Moerbeke case was started: a spectral curve, a discriminant set and a bifurcation diagram of the moment map are explicitly shown, and characteristic exponents for determining the type of critical points of rank $0$ and $1$ of the moment map are presented. In this paper we present an algorithm for constructing Liouville tori. Examples are given of bifurcations of Liouville tori at the intersection of bifurcation curves for reconstructions of one torus into two tori and of two tori into two tori.

Keywords: integrable hamiltonian systems, bifurcation diagram, bifurcations of Liouville tori.

Funding Agency Grant Number
Russian Foundation for Basic Research 16-01-00170_а
16-01-00809_а


DOI: https://doi.org/10.20537/vm170404

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Bibliographic databases:

Document Type: Article
UDC: 517.938.5, 531.38
MSC: 70E05, 70E17, 37J35
Received: 04.12.2017

Citation: S. V. Sokolov, “The Adler–van Moerbeke integrable case. Visualization of bifurcations of Liouville tori”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 27:4 (2017), 532–539

Citation in format AMSBIB
\Bibitem{Sok17}
\by S.~V.~Sokolov
\paper The Adler--van Moerbeke integrable case. Visualization of bifurcations of Liouville tori
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2017
\vol 27
\issue 4
\pages 532--539
\mathnet{http://mi.mathnet.ru/vuu606}
\crossref{https://doi.org/10.20537/vm170404}
\elib{http://elibrary.ru/item.asp?id=32248455}


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  • Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
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