Mathematical Methods in Science and Engineering, 2nd Edition

Chapter 16Variational Analysis

The variational analysis is basically the study of changes. We are often interested in how a system will react to small changes in its parameters. Variational analysis constitutes a powerful tool for determining in which direction a process will go. It is for this reason that it has found a wide range of applications not just in physics and engineering but also in financial mathematics and economics. In applications, we frequently encounter cases where the desired quantity is the one that extremizes a certain integral. Compared to ordinary calculus, where one deals with functions of numbers, these integrals are functions of some unknown function and its derivatives, hence they are called functionals. Search for the extremum of a function yields the points at which the function takes its extremum values. In the case of functionals, the variational analysis gives a differential equation that needs to be solved for the desired function that makes the functional an extremum. After Newton's formulation of mechanics, Lagrange developed a new formalism where the equations of motion are obtained from a variational integral called the action. This new formulation made applications of Newton's theory to many-body problems and continuous systems possible. Today, in making the transition to quantum mechanics and to quantum field theories, a Lagrangian formulation is a must. Geodesics are the shortest paths between two points in a given space and constitute ...