Thecalculusofvariations Lagrangianmechanics and Optimal Control

The calculus of variations addresses the need to optimize certain quantities over sets of functions. We begin by defining the notion of a functional and its role in performing calculus on variations of curves. In order to identify those functions which are extremals of a functional, we establish the first variation and the Euler-Lagrange Equations. A formulation of the second variation follows, providing a means of identifying different types of extremals. Both the first and second variations are illustrated using several examples. We then introduce the calculus of variations as it applies to classical mechanics, resulting in the Principle of Stationary Action, from which we develop the foundations of Lagrangianmechanics. Finally, we examine an extension of the calculus of variations in optimal control. We concludewith Pontryagin’sMaximumPrinciple and its applications in control theory.