2.3 First-order necessary conditions for weak extrema

In this section we will derive the most fundamental result in calculus of variations: the Euler-Lagrange equation. Unless stated otherwise, we will be working with the Basic Calculus of Variations Problem defined in Section 2.2. Thus our function space $V$ is $\mathcal C^1([a,b],\mathbb{R})$ , the subset $A$ consists of functions $y\in V$ satisfying the boundary conditions (2.8), and the functional $J$ to be minimized takes the form (2.9). The Euler-Lagrange equation provides a more explicit characterization of the first-order necessary condition (1.37) for this situation.

In deriving the Euler-Lagrange equation, we will follow the basic variational approach presented in Section 1.3.2 and consider nearby curves of the form

$$y+\alpha \eta$$ (2.10)

where the perturbation $\eta:[a,b]\to \mathbb{R}$ is another $\mathcal C^1$ curve and $\alpha$ varies in an interval around 0 in $\mathbb{R}$ . For $\alpha$ close to 0, these perturbed curves are close to $y$ in the sense of the 1-norm. For this reason, the resulting first-order necessary condition handles weak extrema. However, since a $\mathcal C^1$ strong extremum is automatically a weak extremum, every necessary condition for the latter is necessary for the former as well. Therefore, the Euler-Lagrange equation will apply to both weak and strong extrema, as long as we insist on working with $\mathcal C^1$ functions. As we know, using the 0-norm allows us to relax the $\mathcal C^1$ requirement, but this will in turn necessitate a different approach (i.e., a refined perturbation family) when developing optimality conditions. We will thus refer to the conditions derived in this section as necessary conditions for weak extrema, in order to distinguish them from sharper conditions to be given later which apply specifically to strong extrema. Similar remarks apply to other necessary conditions to be derived in this chapter. The sufficient condition of Section 2.6.2, on the other hand, will apply to weak minima only.