Theory of Tensor Invariants of Integrable Hamiltonian Systems. II. Theorem on Symmetries and Its Applications

Abstract

The theorem on symmetries is proved that states that a Liouville-integrable Hamiltonian system is non-degene\-rate in Kolmogorov's sense and has compact invariant submanifolds if and only if the corresponding Lie algebra of symmetries \(\) is abelian. The theorem on symmetries has applications to the characterization problem, to the integrable hierarchies problem, to the necessary conditions for the strong dynamical compatibility problem, and to the problem on master symmetries. The invariant necessary conditions for the non-degenerate C-integrability in Kolmogorov's sense of a given dynamical system V are derived. It is proved that the C-integrable Hamiltonian system is non-degenerate in the iso-energetic sense if and only if the corresponding Lie algebra of the iso-energetic conformal symmetries \(\) is abelian. An extended concept of integrability of Hamiltonian systems on the symplectic manifolds Mn, n= 2k, is introduced. The concept of integrability describes the Hamiltonian systems that have quasi-periodic dynamics on tori \(\) or on toroidal cylinders \(\) of an arbitrary dimension \(\). This concept includes, as a particular case, all Hamiltonian systems that are integrable in Liouville's classical sense, for which \(\). The A-B-C-cohomologies are introduced for dynamical systems on smooth manifolds.