On the Lagrangian and Hamiltonian Aspects of Infinite-Dimensional Dynamical Systems and their Finite-Dimensional Reductions

Abstract

Some aspects of the description of Lagrangian and Hamiltonian formalisms naturally arising from the invariance structure of given nonlinear dynamical systems on an infinite-dimensional functional manifold are presented. The basic ideas used to formulate the canonical symplectic structure are borrowed from the Cartan theory of differential systems on associated jet-manifolds. The symmetry structure reduced on the invariant submanifolds of critical points of some nonlocal Euler-Lagrange functional is described thoroughly for both differential and differential discrete dynamical systems. The Hamiltonian representation for a hierarchy of Lax-type equations on a dual space to the Lie algebra of integro-differential operators with matrix coefficients, extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems, is obtained via some special Backlund transformation. A relationship between this hierarchy and Lax-integrable spatially two-dimensional systems is studied.