ABSTRACT
The basic problem in the calculus of variations is to determine a function such that a certain functional, often an integral involving that function and certain of its derivatives, takes on maximum or minimum values. The elementary part of the theory is concerned with a necessary condition that the required function must satisfy. To show mathematically that the function obtained actually maximizes or minimizes the integral is much more difficult than the corresponding problems of the differential calculus. Variational problems are easily derived from the differential equation and associated boundary conditions by multiplying by the variation and integrating the appropriate number of times. The variations in some cases cannot be arbitrarily assigned because of one or more auxiliary conditions that are usually called constraints.
