  In this paper we discuss a general framework based on symplectic geometry for
the study of second order conditions in constrained variational problems on
curves. Using the notion of L-derivatives we construct Jacobi curves, which
represent a generalization of Jacobi fields from the classical calculus of
variations, but which also works for non-smooth extremals. This construction
includes in particular the previously known constructions for specific types of
extremals. We state and prove Morse-type theorems that connect the negative
inertia index of the Hessian of the problem to some symplectic invariants of
Jacobi curves.

Optimization and Control;Differential Geometry