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On genericity of integrable Hamiltonian systems of Bott type. (English. Russian original) Zbl 0830.58014
Russ. Acad. Sci., Sb., Math. 81, No. 1, 87-99 (1995); translation from Mat. Sb. 185, No. 1, 107-120 (1994).
A first integral \(F\) of the Hamiltonian system \(v =s \text{grad} H\) is said to be Bott on \(Q_h = \{x \in M^4 : H(x)=h\}\) if the critical set of \(F\) on \(Q_h\) is a disjoint union of nondegenerate manifolds. The system \(v =s \text{grad} H\) is said to be of Bott type on \(Q_h\) if it admits a Bott integral.
The author shows that the set of all systems of Bott type is of first category in the set of all integrable systems in the weak metric. He also shows that the Bott type systems form an open set in the strong metric which it is dense in the set of systems satisfying an additional condition concerning the critical set of \(F\).
MSC:
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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