Prykarpatsky Y. Investigation of algebraic-analytical and topological-geometrical properties of the integrable dynamical systems and their adiabatic perturbations

A symplectic approach for partially solving the problem of algebraic-analytical construction of integral submanifold mappings for integrable via the nonabelian Liouville-Arnold theorems for Hamiltonian systems on canonically symplectic phase spaces is developed. The existence problem of a global set of commuting invariants on the whole phase space was analized. The differential-geometric structures related with adiabatically perturbed Hamiltonian systems on symplectic manifolds are studied. A Hamiltonian connections on the corresponding fiber bundle with a compatible Lie group action is constructed, the adiabatic property of the related momentum mapping is stated. A new approach to Melnikov-Samoilenko stability problem of the invariant torus of the adiabatically perturbed completely integrable oscillatory Hamiltonian systems is devised. Special "virtual" canonical transformations of the phase space in the vicinity of the toroidal integral submanifold by means of separable the Hamilton-Jacobi variablesand algebraic solutions to the associated Picard-Fucks type equations are proposed. The stability problem is reduced to that expressed in the canonical Bogoliubov form. A J. Mather's approach to studying ergodic measures of nonautonomous periodic Hamiltonians flows on exact symplectic manifolds having applications in many problems of mechanics and mathematical physics is developed. Based on homology properties of invariant probability measures minimizing specially constructed Lagrangian functionals, and on symplectic theory developed by A. Floer and others for studing of symplectic actions and related transversal splittings of Lagrangian manifolds, the Mather's type -function related with suitable of ergodic measures on invariant manifolds of nonautonomous Hamiltonian systems is proposed. The symplectic structures related with Lagrangian and Hamiltonian formalisms structures naturally arising from the invariance properties of given nonlinear dynamical systems on infinitedimensional functional manifolds are studied. The symmetry properties of dynamical systems invariantly reduced upon critical points of nonlocal Euler-Lagrange functionals are described thoroughly for both functional and differential-difference dynamical systems. The Hamiltonian representation for a hierarchy of Lax type equations on a dual space to the Lie algebra of integral-differential operators with matrix coefficients, extended by the eigenfunctions evolution of the corresponding spectral problems, is obtained making use of a specially constructed Backlund type transformation. Its connection with Lax type integrable spatially two-dimensional systems is stated. Differential-geometric and topological structures of special de Rham-Hodge complexes related with Delsarte-Lions transmutations of multi-dimensional differential operators in Hilbert spaces are studied. Based on the specially defined de Rham-Hodge-Skrypnik differential complexes the relationships with spectral theory and special Berezansky type congruence properties of Delsarte transmuted operators are stated. Some applications to multidimensional differential operators are done including three-dimensional Laplace operator, two-dimensional Dirac operator and its multidimensional affine extension, related with self-dual Yang-Mills equations. Key words: dynamical system, Liouville-Arnold integrability, integral manifold, imbedding mapping, adiabatic invariants, ergodic measure, connection, Lagrangean functional, reduction, Delsarte-Lions transformation.