Geometric Aspects of the Lame Equation and Plate Theory

Over the past few years, it is gradually understood that de Rham Cohomology Theory is closely related to Saint-Venant's compatibility condition in the Elasticity Theory. In this article, we will discuss the Hodge Theory and de Rham Cohomology Theory hidden in the Lame Equation and in the Plate Theory. we will prove a Decomposition Theorem for the solutions of the Lame Equation, and then use this Decomposition Theorem to present a modified Plate Theory, which is compatible with the general principles of Physics and the mathematical structure of the Lame Equation.