Completeness Of The Eigenfunction Systems Of Infinite Dimensional Hamiltonian Operators And Its Applications In Elasticity

Combining the elasticity with infinite dimensional Hamiltonian operators, Academician Zhong has proposed the method of separation of variables based on Hamiltonian systems, and a new systematic methodology for the theory of elasticity is further established. The theoretical basis of this method is the com-pleteness of the eigenfunction systems (symplectic orthogonal systems) of the infinite dimensional Hamiltonian operators. In this dissertation, centering on the completeness of the eigenfunction systems of the infinite dimensional Hamiltonian operators, some research works have been done as follows:1. The completeness of the eigenfunction system of the infinite dimensional Hamiltonian operators, in the sense of Cauchy's principal value, is studied;2. The problem of invertibility of the infinite dimensional Hamiltonian operators is studied;3. The problem that the infinite dimensional Hamiltonian operators generate semigroups is studied;4. As a supplement of the method of separation of variables based on Hamilto-nian systems, the eigenfunctions expansion method based on upper triangular operator matrices is also studied.Firstly, the eigenfunction system of infinite dimensional Hamiltonian op-erators appearing in the bending problem of rectangular plate with two oppo-sites simply supported is studied. In the sense of Cauchy's principal value, the completeness of the extended eigenfunction system is proved. Then the incom-pleteness of the extended eigenfunction system in general sense is proved. So the completeness of the symplectic orthogonal system of the infinite-dimensional Hamiltonian operator for this kind of plate bending equation is proved. And then a theorem is presented, which the general solution of the infinite dimen-sional Hamiltonian system is equivalent to the solution function system series expansion. It gives to theoretical basis of the methods of separation of vari-ables based on Hamiltonian system for this kind of equations. Secondly, the eigenfunction system of the infinite dimensional Hamiltonian operator appear- ing in the rectangular plates with one side simply supported and the opposite side slidingly supported is studied. In the sense of Cauchy's principal value, the completeness of the extended eigenfunction system is proved. It is provided the feasibility to solve the plane elasticity problem by the symplectic eigenfunction expansion method. Then the general solutions for the plane elasticity problem is derived. Furthermore, it is indicated what boundary conditions for the plane elasticity problem can be solved by this method. Finally, the problem of the free vibration of rectangular Kirchhoff plates with two opposite edges simply supported is studied. The governing differential equations for free vibration of rectangular Kirchhoff plates is rewritten as a Hamiltonian system based on the known results, and the associated Hamiltonian operator is obtained. Then, in the sense of Cauchy's principal value, the completeness of the eigenfunction system is proved. And then the exact general solution for the corresponding Hamilto-nian system of the problem is derived by calculating. Examples are given to illustrate that, combining the general solution with the corresponding boundary conditions, the frequency equations and the transverse displacement functions for Levy-type plates can be derived. Furthermore, it is indicated what boundary conditions for the plane elasticity problem can be solved by this method.In course of study the completeness of the eigenfunction systems of the in-finite dimensional Hamiltonian operators, we discover that the completeness of many infinite dimensional Hamiltonian operators is related to the invertibility of the corresponding Hamiltonian operators. In order to explore the intrinsic rela-tionship between them and taking into account the importance of the invertibility of Hamiltonian operators, we studied the invertibility of Hamiltonian operators. In this paper, taking full advantage of structure of infinite dimensional Hamilto-nian operators, the invertibility and distribution of Point spectrum of a class of Hamiltonian operators are obtained.Many practical problems have been solved by eigenfunction expansion method based on Hamiltonian systems. However we discover that there are some infinite dimensional Hamiltonian operators whose eigenfunction systems is incomplete-ness during the study of the completeness. In order to solve this kind of problems, which can not be solved by the method of eigenfunction expansion method based on Hamiltonian systems, we need to seek other methods. Based on above rea-sons, two methods are adopted in this paper. One is semigroup methods in the framework of the infinite dimensional Hamiltonian system, and the other is eigen-functions expansion methods based on upper triangular operator matrix, which is out of the framework of the infinite dimensional Hamiltonian system. As for the problem that infinite dimensional Hamiltonian operators generate semigroups, applying Hille-Yosida theorem, the infinite dimensional Hamiltonian operators is researched. Then it is proved that the existence theorem of solutions for a class of initial value problems of infinite dimensional Hamiltonian systems. Par-ticularly, the obtained result is applied to the infinite dimensional Hamiltonian system generated by a kind of hyperbolic partial differential equations and the existence theorem of solutions is obtained. For the eigenfunctions expansion method based on upper triangular operator matrix, the two-dimensional (2D) elasticity problems based on stress formulation is studied. The fundamental sys-tem of partial differential equations of the2D problems is rewritten as an upper triangular differential system based on the known results, and then the associ-ated operator matrix is obtained. The two simpler complete orthogonal systems of eigenfunctions are obtained, which belong to the two block operators arising in the operator matrix, respectively. Then a more simple and convenient general solution for the2D problem is derived by the eigenfunction expansion method. Furthermore, it is indicated what boundary conditions for the2D problem can be solved by this method.In this dissertation, the research works provide some theoretical foundations for solving the practical problems based on Hamiltonian system. And the research works are useful for further research on infinite dimensional Hamiltonian systems and eigenfunction expansion methods.