  The generalization of the Maupertuis principle to second-order Variational
Calculus is performed. The stability of the solutions of a natural dynamical
system is thus analyzed via the extension of the Theorem of Jacobi. It is shown
that the Morse Theory of the trajectories in the dynamical system is identical
to the Morse Theory of geodesics in the Jacobi metric, even though the
second-variation functionals around the action and the Jacobi length do not
coincide. As a representative example, we apply this result to the study of the
separatrix solutions of the Garnier System.
