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| Method of Separation of Variables and Hamiltonian System |
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| DOI：10.7511/jslx19913033 |
| KeyWord:seperation of variables,Hamilton system,simplectic,Hamiltonian matrix,Eigen-function expansion. |
| Zhong Wanxie |
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| Abstract: |
| 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. |
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