2.6 Second-order conditions

In Section 2.3 we used the first variation to obtain the first-order necessary condition for optimality expressed by the Euler-Lagrange equation. In this section we will work with the second variation and derive first a necessary condition and then a sufficient condition for optimality. The setting is that of the Basic Calculus of Variations Problem and weak minima (cf. the discussion at the beginning of Section 2.3).

Recall from Section 1.3.3 the second-order expansion

which defines the quadratic form called the second variation. The basic second-order necessary condition (1.40) says that if a curve is a local minimum of , then for all admissible perturbations we must have . In the present context, the function space and the subset are as at the beginning of Section 2.3, and so admissible perturbations are functions satisfying (2.11). Using the fact that the functional takes the form (2.9), we will derive in Section 2.6.1 a more explicit second-order necessary condition for this situation.

We also discussed in Section 1.3.3 that to develop
a second-order *sufficient* condition for a local minimum, we need
to assume the first-order necessary condition,
strengthen the second-order necessary condition
to
, and then try to prove that the second-order
term dominates the higher-order term in the expansion (2.56).
For the variational problem at hand, we will be able to carry out this program
in Section 2.6.2.

Of course, second-order conditions for a local *maximum* are easily obtained
by reversing the inequalities (or, equivalently, by replacing
with
).
For this reason, we confine our attention to minima.

- 2.6.1 Legendre's necessary condition for a weak minimum
- 2.6.2 Sufficient condition for a weak minimum