On typical topological properties of integrable Hamiltonian systems. (Russian) Zbl 0647.58018
The non-singular surfaces of constant energy associated to the integrable Hamiltonian systems defined on four-dimensional symplectic manifolds are studied. It is shown that the irreducible surfaces of constant energy corresponding to such integrable systems are completely defined by their fundamental groups. For the Hamiltonian systems it is proven that the surfaces of constant energy are irreducible.
It is established that all closed periodic solutions of the Hamiltonian systems are non-homotopical null and the circles corresponding to the maximum or minimum values of the Bott integral are non-homotopical to each other. It is pointed out that the results on topological integrability are important in the study of Liouville torus bifurcations.
It is established that all closed periodic solutions of the Hamiltonian systems are non-homotopical null and the circles corresponding to the maximum or minimum values of the Bott integral are non-homotopical to each other. It is pointed out that the results on topological integrability are important in the study of Liouville torus bifurcations.
Reviewer: G.Zet
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.) |