3. Direct methods

On first reading, it may seem that the classical methods presented in provide a sufficiently flexible framework for dealing with the problems of calculating variations. However, a careful reading reveals three important weaknesses of these methods:

• we have never shown that the problem under consideration has a solution. We therefore have an existence problem;

• Even if the existence question were solved, the necessary conditions to be satisfied by the minimizer presuppose that we know how to solve differential equations such as the Euler-Lagrange equation, the associated Hamiltonian system or the Hamilton-Jacobi equation. But this is generally a very difficult problem. In fact, modern methods do exactly the opposite. We show that the variational problem admits a minimizer and, in this way,...