Symplectic Subspace Research Articles

Symplectic solution system for reissner plate bending

Based on the Hellinger-Reissner variatonal principle for Reissner plate bending and introducing dual variables, Hamiltonian dual equations for Reissner plate bending were presented. Therefore Hamiltonian solution system can also be applied to Reissner plate bending problem, and the transformation from Euclidian space to symplectic space and from Lagrangian system to Hamiltonian system was realized. So in the symplectic space which consists of the original variables and their dual variables, the problem can be solved via effective mathematical physics methods such as the method of separation of variables and eigenfunction-vector expansion. All the eigensolutions and Jordan canonical form eigensolutions for zero eigenvalue of the Hamiltonian operator matrix are solved in detail, and their physical meanings are showed clearly. The adjoint symplectic orthonormal relation of the eigenfunction vectors for zero eigenvalue are formed. It is showed that the all eigensolutions for zero eigenvalue are basic solutions of the Saint-Venant problem and they form a perfect symplectic subspace for zero eigenvalue. And the eigensolutions for nonzero eigenvalue are covered by the Saint-Venant theorem. The symplectic solution method is not the same as the classical semi-inverse method and breaks through the limit of the traditional semi-inverse solution. The symplectic solution method will have vast application.