Abstract
The paper is devoted to a short explanation of the topological classification (up to Liouville equivalence) of the integrable geodesic flows of two-dimensional surfaces of revolution with potential. The classification is given in the terms of so-called “marked molecules,” i.e., Fomenko–Zieschang invariants for integrable systems with two degrees of freedom on three-dimensional isoenergy sufraces.
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References
Fomenko, A.T.: Symplectic Geometry, Second revised edn. Gordon and Breach, New York (1995)
Bolsinov, A.V., Fomenko, A.T.: Integrable Geodesic Flows on Two-Dimensional Surfaces. Consultants Bureau, New York (2000). (Kluwer Academic/Plenum Publishers, New York)
Fomenko, A.T., Konyaev, A.Y.: Algebra and geometry through Hamiltonian systems. In: Zgurovsky, M.Z., Sadovnichiy, V.A. (eds.) Continuous and Distributed Systems. Theory and Applications. Solid Mechanics and Its Applications, pp. 3–21. Springer, Berlin (2014)
Bolsinov, A.V., Fomenko, A.T.: Integrable Hamiltonian Systems: Geometry, Topology, Classification. Chapman and Hall/CRC, (A CRC Press Company) Boca Raton (2004)
Bolsinov, A.V., Fomenko, A.T.: Integrable Hamiltonian Systems: Geometry, Topology, Classification, 2. Chapman and Hall/CRC, (A CRC Press Company) Boca Raton
Fomenko., A.T.: A bordism theory for integrable nondegenerate Hamiltonian systems with two degrees of freedom. A new topological invariant of higher-dimensional integrable systems. Math. USSR Izvestiya 39(1), 731–759 (1992)
Bolsinov, A.V., Fomenko, A.T.: Integrable geodesic flows on the sphere, generated by Goryachev-Chaplygin and Kowalewskaya systems in the dynamics of a rigid body. Math. Notes 56(1–2), 859–861 (1994)
Kudryavtseva, E.A., Nikonov, I.M., Fomenko, A.T.: Maximally symmetric cell decompositions of surfaces and their coverings. Sb. Math. 199(9), 3–96 (2008)
Fomenko, A.T., Konyaev, AYu.: New approach to symmetries and singularities in integrable Hamiltonian systems. Topol. Appl. 159, 1964–1975 (2012)
Fomenko, A.T.: Hidden symmetries in integrable Hamiltonian systems. In: Progress in Analysis. Proceedings of the 8th Congress of the International Society for Analysis, its Applications, and Computation. pp. 22–27 August 2011. Moscow, Peoples’ Friendship University of Russia, 2012. Vol. 1, 26–41
Kudryavtseva, E.A., Fomenko, A.T.: Every finite group is a symmetry group of a map (of atom-bifurcation), Vestnik MGU, 1. Matem. Mech. 3, 21–29 (2013)
Fomenko, A.T., Kudryavtseva, E.A.: Each finite group is a symmetry group of some map (an atom-bifurcation). Mosc. Univ. Math. Bull. 68(3), 148–155 (2013)
Kantonistova, E.O.: Integer lattices of action variables for generalized Lagrange case. Mosc. Univ. Math. Bull. 1, 54–58 (2012). ISSN 0201-7385
Kantonistova, E.O.: Integer lattices of action variables for spherical pendulum system. Mosc. Univ. Math. Bull. 69(4), 135–147 (2014)
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Fomenko, A.T., Kantonistova, E.O. (2015). Topological Classification of Geodesic Flows on Revolution 2-Surfaces with Potential. In: Sadovnichiy, V., Zgurovsky, M. (eds) Continuous and Distributed Systems II. Studies in Systems, Decision and Control, vol 30. Springer, Cham. https://doi.org/10.1007/978-3-319-19075-4_2
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DOI: https://doi.org/10.1007/978-3-319-19075-4_2
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