ABSTRACT
The fundamental problem of the calculus of variations involved the first derivative of the unknown function. This chapter discusses the presence of higher order derivatives. Clearly the solution of this may be achieved by classical calculus tools with four boundary conditions. It is also common in engineering applications to impose boundary conditions on some of the derivatives (Neumann boundary conditions). Derivative constraints may also be applied to the case of higher order derivatives. Hence, the solution may be obtained by the application of a system of two Euler-Poisson equations. It is very common in engineering practice that the highest derivative of interest is of second order. Accelerations in engineering analysis of motion, curvature in description of space curves, and other important application concepts are tied to the second derivative. Depending on the particular application circumstance, the linear system of Euler-Lagrange equations may be more conveniently solved than the quadratic Euler-Poisson equation.
