Variable End Point Problem

The variational derivative of a functional J[y] can be defined as δJ/δy = F y (x, y, y ′)− d dx F y ′ (x, y, y ′) [1, pp. 27–29]. Euler's equation essentially states that the variational derivative of the functional must vanish at an extremum. This is analogous to the well-known result from calculus that the derivative of a function must vanish at an extremum. In this section, we consider a simple case of the variable end point problem, which is stated as follows: Among all curves whose end points lie on two vertical lines x = a and x = b, find the curve for which the functional J[y] = b a F (x, y, y ′) dx (1) has an extremum. We determine the variation of the functional (1), which is the linear component of the increment