We begin this chapter by describing a simple fixed endpoint discrete variational problem. We initially consider fixed step sizes of length 1. However, in Section 4.7 we will let the step sizes be of variable length. Assume f(t, y, r) for each t in the discrete interval [a + 1, b + 2] is of class C 2 with respect to the components of the n dimensional vector variables y and r. We define a set of admissible functions by $$\mathcal{F} = \{ y:[a,b + 2] \to {{R}^{n}}:y(a) = \alpha ,\quad y(b + 2) = \beta \}$$ where α and β are given column n vectors.