Completeness Of Eigenfunction System Of Two Kinds Of Hamiltonian Operators And Its Application To Symplectic Elasticity

The necessary and sufficient conditions for the completeness of the eigenfunction system of a class of 2 × 2 Hamiltonian operators in the sense of Cauchy principal value are obtained,and the new completeness theorem is applied to 4 x 4 infinite dimensional Hamiltonian operator matrices.Based on the characteristics of the separable Hamiltonian system,a new method of separation of variables is obtained.As an application of the theorem,the bending problem of rectangular cantilever thin plate is studied,and the general solution of the equation is given,which proves the validity of the new method.The plate and shell problems play an important role in the study of elastic mechanics model.By introducing the state parameters the basic equations of rectangular moderate-ly thick plate are transformed into the form of infinite dimensional separable Hamiltonian.Based on the biorthogonal relationship of eigenfunction system,the complete biorthog-onal expansion theorem of Hamiltonion canonical equation for rectangular moderately thick plate is obtained,and the Fourier series solution of rec tangular moderately thick plate under the two opposite simply supported boundary condition is obtained.