ABSTRACT
This chapter discusses the calculus of variations and the associated Euler-Lagrange equation for the case of a single variable and the case of multi- independent and dependent variables. The origin of the calculus of variations can be traced back to the time of Bernoulli and Euler. The application of the calculus of variations is mainly used in searching an optimum solution of problems. The study of the existence of an extremal value of functionals results in the so-called direct method of the calculus of variation. There is close resemblance between differential calculus and the calculus of variations. The brachistochrone was first formulated by Galileo in 1638, but he was unable to solve the problem. This problem was considered one of the founding problems in the calculus of variations. The calculus of variations is a classical mathematical topic in applied mathematics and engineering.
