关于infinite -dimensional动力系统的Lagrange和Hamilton性质及其有限维约化
On the Lagrangian and Hamiltonian aspects of infinite -dimensional dynamical systems and their finite-dimensional reductions
摘要：
本文给出了由无穷维泛函流形上的非线性动力系统的不变性结构所自然产生的拉格朗日和哈密顿形式。用于制定规范的辛结构的基本思想是借用从嘉当理论的微分系统相关的射流-流形。对微分和微分离散动力系统，给出了在非局部Euler-Lagrange泛函临界点的不变子流形上约化的对称结构.通过特殊的Backlund变换，得到了矩阵系数积分-微分算子李代数的对偶空间上一族Lax型方程的Hamilton表示，并将其推广到相应谱问题的本征函数和伴随本征函数的演化上.研究了该族与二维Lax可积系统的关系。
英文摘要：
A description of Lagrangian and Hamiltonian formalisms naturally arisen from the invariance structure of given nonlinear dynamical systems on the infinite--dimensional functional manifold is presented. The basic ideas used to formulate the canonical symplectic structure are borrowed from the Cartan's 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 integral-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. The connection of this hierarchy with integrable by Lax spatially two-dimensional systems is studied.
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