Saddle point techniques

From the beginning of the calculus of variations over 300 years ago, scientists were interested in finding extrema of expressions (now called functionals) involving unknown functions. It was discovered that functions producing extrema are solutions of differential equations (called Euler equations). In one dimentional problems these were ordinary differential equations, and the easiest procedure was to solve the corresponding Euler equation and then check if the solution produces the desired maximum or minimum. However, as researchers began to study higher dimensional problems, it became obvious that it was more difficult to solve the Euler equations than obtain an extremum of the functional. Moreover, many partial differential equations and systems that were being studied by various scientists happened to be the Euler equations of functionals. In this case, any critical point of the corresponding functional will provide a solution of the given equation. This led to the search for critical points of functionals. In this paper we describe several methods of finding critical points. We present applications.