Extrema and Variational Calculus
The extreme values seem to say little more than where the local maxima and minima of a function are, or where the highest and lowest points on a mountain range might be. This chapter discusses the principles leading to applications of the calculus of variations in physics. A similar analysis of extrema can be carried out for functions of several variables. A point that is not a local extremum is a saddle point. Interest in extrema problems in classical mechanics began near the end of the seventeenth century with Newton and Leibniz. The chapter explores some standard examples leading to finding the extrema of functionals. In order to find the geodesic equation, the Variational Principle is used which states that freely falling test particles follow a path between two fixed points in spacetime that extremizes the proper time.