Kalashnikov, V. V. Nondegenerate systems and generic properties of the integrable Hamiltonian systems. (English. Russian original) Zbl 0936.37026 J. Math. Sci., New York 94, No. 4, 1558-1563 (1999); translation from Zap. Nauchn. Semin. POMI 235, 184-192 (1996). In the paper the author clarifies whether the nondegenerate systems are generic among all integrable systems. This question is important because the nondegenerate systems can be described up to topological equivalence by the Fomenko-Zieschang isoenergetic invariant. It has been proved that every integrable nondegenerate Hamiltonian system can be made integrable degenerate on the given energetic level by a small perturbation in the weak metric, i.e. degenerate systems are dense in this topology. Besides under the assumptions that a Hamiltonian system has a nondegenerate integral \(f\) on \(Q_{h}=\{H=h\}\), all critical manifolds of \(f|_{Q_{h}}\) are circles and there is only one circle on every critical level of \(f\), the author proves that the Fomenko-Zieschang isoenergetic invariant of the system on \(Q_{h}\) does not change under small perturbations in the strong metric. Reviewer: Alexei Ivanov (St.Peterburg) MSC: 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry Keywords:integrable systems; Fomenko-Zieschang topological invariant; isoenergetic invariants; Hamiltonian system × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] V. I. Arnold,Geometrical Methods in the Theory of Ordinary Differential Equation, Springer (1988). [2] A. V. Bolsinov, ”Methods of calculation of the Fomenko-Ziechang invariant,”Adv. Sov. Math.,6, 147–183 (1991). · Zbl 0744.58029 [3] A. V. Bolsinov, A. T. Fomenko, and S. V. Matveev, ”Topological classification of integrable systems with two degrees of freedom. List of systems of small complexity,”Usp..Mat. Nauk,45 No. 2 (1990). · Zbl 0696.58019 [4] A. T. Fomenko, ”The topology of surfaces of constant energy in integrable Hamiltonian systems and obstruction to integrability,”Izv. Akad. Nauk SSSR, Ser. Mat.,50, 1276–1307 (1986). · Zbl 0619.58023 [5] A. T. Fomenko,Symplectic geometry. Methods and applications [in Russian], Izdat. Moskov. Univ., Moscow (1988). · Zbl 0751.53002 [6] V. V. Kalashnikov, ”On the typicalness of Bottian integrable Hamiltonian systems,”Mat. Sb.,185, No. 1, 107–120 (1994). [7] H. Knörrer, ”Singular fibers of the momentum mapping for integrable hamiltonian systems,”J. Reine und Ang. Math.,355, 68–107 (1985). · Zbl 0542.58005 [8] Nguen Tien Zung, ”On the general position property of simple Bott integrals,”Usp. Mat. Nauk,45, No. 4, 161–162 (1990). · Zbl 0724.58030 [9] A. A. Oshemkov, ”Fomenko invariants for the main integrable cases of the rigid body motion equations,”Adv. Sov. Math.,6, 67–146 (1991). · Zbl 0745.58028 [10] G. P. Paternain, ”On the topology of manifolds with completely integrable geodesic flows,”Ergod. Th. Dyn. Syst.,12, 109–121 (1992). · doi:10.1017/S0143385700006623 [11] E. N. Selivanova, ”Topological classification of integrable Bott geodesic flows on two-dimensional torus,”Adv. Sov. Math.,6, 209–228 (1991). · Zbl 0744.58034 [12] N. Weaver, ”Pointwise periodic homeomorphisms of continua,”Ann. Math.,95, 83–85 (1972). · Zbl 0231.58010 · doi:10.2307/1970855 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.