Applied Calculus of Variations for Engineers, Third edition, 3rd Edition

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Higher order derivatives

The fundamental problem of calculus of variations involved the first derivative of the unknown function. In this chapter, we will allow the presence of higher order derivatives that lead to the so-called Euler-Poisson equation. The chapter will also present a method applying an algebraic constraint on the derivative. Finally the technique of linearization of second order problems will be discussed and illustrated.

 

4.1The Euler–Poisson equation

First let us consider the variational problem of a functional with a single function, but containing its higher derivatives:

I(y)=∫x0x1f(x,y,y′,…,y(m))dx.

Accordingly, boundary conditions for all derivatives will also be given as

y(x0)=y0,y(x1)=y1,

y′(x0)=y0′,y′(x1)=