Variational Methods

This chapter begins with a review of the underlying theory of variational methods. One of the first applications of classical differential calculus that a student encounters is the characterization of extrema of real valued functions. The chapter reviews Hamilton’s principle as it is applied to linearly elastic systems, and presents Hamilton’s principle for linear piezoelectricity. Hamilton’s Principle is used to determine the equations of motion of a mechanical system from the stationarity of the action integral. One popular example of the application of Hamilton’s principle that is used to illustrate the application of variational methods in vibrations or structural dynamics considers a linearly elastic Bernoulli–Euler beam. The chapter shows that Hamilton’s Principle provides an effective alternative to Newton’s formulation for deriving the equations of motion of linearly elastic continua. It is important to note that variational formulations of electromechanical systems have been studied extensively and documented in numerous sources.