An inverse problem from sub-Riemannian geometry

The geodesics for a sub-Riemannian metric on a three-dimensional contact manifold M form a 1-parameter family of curves along each contact direction. However, a collection of such contact curves on M, locally equivalent to the solutions of a fourth-order ODE, are the geodesics of a sub-Riemannian metric only if a sequence of invariants vanish. The first of these, which was first identified by Fels, determines if the differential equation is variational. The next two determine if there is a well-defined metric on M and if the given paths are its geodesics.