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## Archive of Issues

Russia Moscow
Year
2017
Volume
27
Issue
4
Pages
532-539
 Section Mathematics Title The Adler-van Moerbeke integrable case. Visualization of bifurcations of Liouville tori Author(-s) Sokolov S.V.a Affiliations Mechanical Engineering Research Institute, Russian Academy of Sciencesa 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 UDC 517.938.5, 531.38 MSC 70E05, 70E17, 37J35 DOI 10.20537/vm170404 Received 4 December 2017 Language Russian Citation Sokolov S.V. The Adler-van Moerbeke integrable case. Visualization of bifurcations of Liouville tori, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. 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The discriminant set and bifurcation diagram of the integrable case of M. Adler and P. van Moerbeke, Nelineinaya Dinamika, 2016, vol. 12, no. 4, pp. 633-650 (in Russian). DOI: 10.20537/nd1604007 Bolsinov A.V., Borisov A.V. Compatible Poisson brackets on Lie algebras, Mathematical Notes, 2002, vol. 72, no. 1, pp. 10-30. DOI: 10.1023/A:1019856702638 Full text