Lesson 1 - Principle of Minimum Potential Energy

Abstract

The purpose of this chapter is to introduce the concept of principle of minimum potential energy and the concept of a functional (a function of a function).The chapter also illustrates the relationship between the potential energy function and equations of statics. The equations of statics that govern the behavior of discrete systems result when the potential energy function is minimized using the procedures of differential calculus. The equilibrium equations for discrete systems are produced when the potential energy function is minimized using the procedures of differential calculus. The principle of minimum potential energy is extended to continuous systems where one can find that the potential energy expression has the form of a functional. The principle of minimum potential energy directly relates the potential energy function and the governing equations of the physical systems being studied. An understanding of this relationship identifies two independent approaches for solving the equilibrium equations. The finite element method finds approximate solutions that directly minimize the potential energy. The finite difference method finds approximate solutions that satisfy the governing equations and indirectly minimize the potential energy. The need to minimize the potential energy functional for continuous systems provides the motivation for the calculus of variations.