×

zbMATH — the first resource for mathematics

On the general position property of simple Bott integrals. (English. Russian original) Zbl 0724.58030
Russ. Math. Surv. 45, No. 4, 179-180 (1990); translation from Usp. Mat. Nauk 45, No. 4(274), 161-162 (1990).
Let (M,\(\omega\)) be a 4-dimensional symplectic smooth manifold, H a Hamiltonian on M, \(v=s \text{grad} H\) the Hamiltonian field and \(Q=\{H=0\}\) a noncritical isoenergetic compact submanifold such that v is Liouville integrable, i.e. there is a function (integral) F on a neighborhood of Q, independent of H and \([F,H]=0\) where [.,.] is the Poisson bracket. The author recalls after Fomenko the notion of the simple and complex Bott integral and sketches the proof of the result stating that, in spite of the fact that the set of all simple Bott integrals is not dense in the set of all Bott integrals, if F is a Bott integral, then there are smooth homotopies \(\omega_ t\), \(F_ t\) (t\(\in [-1,1]\), \(\omega_ 0=\omega\), \(F_ 0=F)\) such that the cohomology classes \([\omega_ t]\) and [\(\omega\) ] are equal and \(F_ t\) is a simple Bott integral, for \(t\neq 0\), of \((M,\omega_ t,H).\)
Several consequences are studied and even-dimensional symplectic manifolds are considered.

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.)
37D15 Morse-Smale systems
PDF BibTeX XML Cite
Full Text: DOI