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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.
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
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