ABSTRACT
This chapter introduces the basic problem in the calculus of variations and discusses some tools and continuity requirements. It seeks to obtain necessary and sufficient conditions for the minimization problem of a real valued function. The chapter provides some types of Euler-Lagrange equations depending on the smoothness of problem, as well as corner conditions and transversality conditions which are necessary conditions for a critical point solution. In addition, there are difficulties about whether problem is well-posed which involve the questions of existence, uniqueness and continuous dependence on the given initial or boundary conditions of the solution. The chapter discusses a hierarchy of four problem types denoted by global minimum, strong relative minimum, weak relative minimum and critical point. The major result is that a critical point solution satisfies the Euler-Lagrange equation, which is a second order ordinary differential equation.
