A complete theory of local minimizers for periodic variational problems

We study the periodic problem in the Calculus of Variations and give a complete characterization of the positive definiteness of the second variation, extending the Reid Roundabout Theorem to periodic boundary conditions. Two formulations are established: one in terms of conjugate and semi-coupled points, and one in terms of the existence of a symmetric solution of the associated Riccati equation satisfying a boundary inequality. We show by an explicit example that the latter condition, while sufficient, is not necessary except in dimension one. As applications, we obtain sufficient conditions for weak and strong local minimizers of periodic solutions of the Euler-Lagrange equation.