ABSTRACT

This chapter generalizes the constraint concept in two senses. There is the possibility of defining the boundary condition at one end of the integral of the variational problem with an algebraic constraint. The first is to allow more difficult, algebraic boundary conditions, and the second is to allow constraints imposed on the interior of the domain as well. Here the authors focus on the simple case of finding the curve of given length between two points in the plane. It is simple to verify that the solution produces the extremum of the original variational problem. A body in a force field is in static equilibrium when its potential energy has a stationary value. This is also known as principle of minimum potential energy. Clearly, depending on the length of the cable between similarly posted suspension locations, different catenary curves may be obtained.