分离变量法与哈密尔顿体系

数学物理与力学中用分离变量法求解偏微分方程经常导致自共轭算子的sturmLiouville问题,在此基础上而得以展开求解。然而在应用中有大量问题并不能导致自共轭算子。本文通过最小势能变分原理,选用状态变量及其对偶变量,导向一般变分原理。利用结构力学与最优控制的模拟理论,导向哈密尔顿体系。将有限维的理论推广到相应的哈密尔顿算子矩阵及共轭辛矩阵代数的理论。拓广了经典的分离变量法,证明了全状态本征函数向量的共轭辛正交归一性质及按本征函数向量展开的理论。以条形板为例,说明了应用。

Method of Separation of Variables and Hamiltonian System

In the theory of mechanics and/or mathematical physics problems in prismatic domain, the method of separation of variables usually leads to the Sturm-Liouville type self-adjoint eigen-problems. However, a number of very important application problems cannot lead to self-adjoint operator for the transverse coordinate. From the minimum potential energy variational principle, by selection of the state and its dual variablts the generalized variational principle is deduced, and then based on the analogy between the theory of structural mechanics and optimal control, the present paper lead the problem to Hamiltonian system. The finite dimensional theory of Hamiltonian system is extended to the corresponding theory of Hamiltonian operator matrix, and adjoint simplectic spaces. The adjoint simplectic ortho-normality relation is proved for the whole state eigenfunction vectors and then the expansion of an arbitrary whole state function vector by the eigenfunction vectors is established. Thus the classical method of separation of variables is extended. The plate bending problem in a strip domain is used for illustration.