Summary
The analytic expression for a Riemannian metric on a 2-sphere, having integrable geodesic flow with an additional integral quadratic in momenta, is given in [Ko1]. We give the topological classification, up to topological equivalence of Liouville foliations, of all such metrics. The classification is computable, and the formula for calculating the complexity of the flow is straightforward. We prove Fomenko's conjecture that, from the point of view of complexity, the integrable geodesic flows with an additional integral linear or quadratic in momenta exhaust “almost all” integrable geodesic flows on the 2-dimensional sphere.
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References
Birkhoff, J. D.Dynamical Systems. M.-L.: OGIZ, (1941).
Bolsinov, A. V., Matveev, S. V., Fomenko, A. T. Topological Classification of Integrable Hamiltonian Systems with Two Degrees of Freedom: A List of the Systems of Small Complexity.Uspekhi Mat. Nauk 45 no. 2, 49–77, (1990) (in Russian).
Fomenko, A. T.Symplectic Geometry: Methods and Applications. Moscow University, 1988.
Fomenko, A. T. Topology of the Constant Energy Surfaces of Integrable Hamiltonian Systems and Obstructions to Integrability.Izv. Akad. Nauk SSSR 50 no. 6, 1276–1307 (1986).
Fomenko, A. T. Topological Classification of All Integrable Hamiltonian Differential Equations of General Type with Two Degrees of Freedom. In T. Ratin, ed.,The Geometry of Hamiltonian Systems. Springer-Verlag (1991).
Fomenkov, A. T. Topological Classification of Integrable Hamiltonian systems. In A. T. Fomenko, ed.,Advances in Soviet Mathematics, American Mathematical Society, vol. 6 (1991).
Fomenko, A. T., Zieschang, H. On Typical Topological Properties of Integrable Hamiltonian Systems.Izv. Akad. Nauk SSSR, Ser. Matem. 52 no. 2, 378–467 (1988).
Kalashnikov, V. V. Geometrical Description of Fomenko's Topological Invariants of Minimax Integrable Hamiltonian Systems on\(\mathbb{S}^3 \), ℝℙ3,\(\mathbb{S}^1 \) ×S 2; and\(\mathbb{T}^3 \). In A. T. Fomenko, ed.,Advances in Soviet Mathematics, American Mathematical Society, vol. 6, 297–304 (1991).
Kolokoltsov, V. N. Geodesic Flows on Two-Dimensional Manifolds with an Additional First Integral Polynomial in Velocities.Izv. Akad. Nauk SSSR, Ser. Matem. 46 no. 5, 994–1010 (1982).
Kolokoltsov, V. N. New Examples of Manifolds with Closed Geodesics.Vestnik. Mosk. Univ., Ser. Matem. Meh., 4, 80–82 (1984).
Kolokoltsov, V. N. Polynomial integrals of geodesic flows on compact surfaces. Ph.D. Diss., Moscow, 1984.
Selivanova, E. N. A Topological Classification of Integrable Bott Geodesic Flows on the Two-Dimensional Torus, in A. T. Fomenko, ed.,Advances in Soviet Mathematics, American Mathematical Society, vol. 6, 209–228 (1991).
Shiffer, M., Spenser, D. K.Functionals on Finite Riemannian Surfaces. M.:IM, 1957.
Nguyen, T. Z., Fomenko, A. T. A Topological Classification of Integrable Bott Hamiltonians on the Isoenergy Three-Dimensional Sphere.Uspekhi Mat. Nauk 45 no. 6, 91–111 (1990).
Nguyen, T. Z. The Complexity of Integrable Hamiltonian Systems on a Given Isoenergy Three-Dimensional Manifold.Mat. sbornik. 183 no. 4, 87–117 (1992).
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Communicated by Jerrold Marsden
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Nguyen, T.Z., Polyakova, L.S. A topological classification of integrable geodesic flows on the two-dimensional sphere with an additional integral quadratic in the momenta. J Nonlinear Sci 3, 85–108 (1993). https://doi.org/10.1007/BF02429860
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DOI: https://doi.org/10.1007/BF02429860