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The purpose of this chapter is to develop an optimization technique known as the calculus of variations. This powerful method can be applied to a variety of mathematical and physical problems to derive the governing differential equation of the problem. Among others, such problems include those that arise from Hamilton's principle and in applying the principle of minimum potential energy to determine the equilibrium configuration of a deformable body. Higher-dimensional problems often lead to some of the classic partial differential equations of mathematical physics.
The calculus of variations deals with the optimization problem of finding an extremal (maxima or minima) of a quantity in the form of an integral. The simplest example of such a problem is to show that the shortest path between two points in space is a straight line. A similar problem formulated from Fermat's principle leads to Snell's law.
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