  It is well-known that fixed energy solutions of a reversible autonomous
Lagrangian system are up to time reparametrization geodesics of the
Jacobi-Maupertuis metric, which degenerates at the boundary of the Hill's
region. In a recent paper, Montgomery proved that geodesics hitting the
boundary at a regular point always contain pairs of focal points, and hence in
particular cannot be minima of the energy functional. Starting from this, we
provide a precise Morse index formula for periodic brake orbits of a reversible
autonomous Lagrangian system by computing the local contribution to the Morse
index provided at each brake instant. We finally discuss an application to a
doubly coupled harmonic oscillator.
