Abstract:
The paper is concerned with the study of the topology of the Liouville foliations of the Dullin-Matveev integrable case. The critical point set of the Hamiltonian is found, the types of isoenergy surfaces are calculated, the non-degeneracy conditions are verified, the types of non-degenerate points of the Poisson action are determined,
the moment map is investigated and the bifurcation diagram is constructed. A test for the Bott property is
verified by numerical simulation. The indices of critical circles, the bifurcation types and the
rough molecules are found. The rough Liouville classification of this integrable case is virtually accomplished as a result.
Bibliography: 24 titles.
\Bibitem{Mos08}
\by A.~Yu.~Moskvin
\paper Topology of the Liouville foliation on a~2-sphere in the Dullin-Matveev integrable case
\jour Sb. Math.
\yr 2008
\vol 199
\issue 3
\pages 411--448
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https://doi.org/10.1070/SM2008v199n03ABEH003926
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This publication is cited in the following 8 articles:
G. P. Palshin, “Topology of the Liouville foliation in the generalized constrained three-vortex problem”, Sb. Math., 215:5 (2024), 667–702
A. T. Fomenko, V. V. Vedyushkina, “Billiards and integrable systems”, Russian Math. Surveys, 78:5 (2023), 881–954
A. T. Fomenko, V. V. Vedyushkina, “Singularities of integrable Liouville systems, reduction of integrals to lower degree and topological billiards: recent results”, Theor. Appl. Mech., 46:1 (2019), 47–63
V. V. Vedyushkina (Fokicheva), A. T. Fomenko, “Integrable geodesic flows on orientable two-dimensional surfaces and topological billiards”, Izv. Math., 83:6 (2019), 1137–1173
Vedyushkina V.V. Fomenko A.T., “Reducing the Degree of Integrals of Hamiltonian Systems By Using Billiards”, Dokl. Math., 99:3 (2019), 266–269
M. P. Kharlamov, P. E. Ryabov, I. I. Kharlamova, “Topological Atlas of the Kovalevskaya–Yehia Gyrostat”, J. Math. Sci. (N. Y.), 227:3 (2017), 241–386
Fomenko A.T., Konyaev A.Yu., “New approach to symmetries and singularities in integrable Hamiltonian systems”, Topology Appl., 159:7 (2012), 1964–1975
M. P. Kharlamov, P. E. Ryabov, “Diagrammy Smeila–Fomenko i grubye invarianty sluchaya Kovalevskoi–Yakhya”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2011, no. 4, 40–59
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