Legendre singularities of sub-Riemannian geodesics associated to Riemannian surfaces

Let $M$ be a surface with a Riemannian metric and $UM$ the unit tangent bundle over $M$ with the canonical contact sub-Riemannian structure $D \subset T(UM)$. In this paper, the complete local classification of singularities, under the Legendre projection $UM \to M$, is given for sub-Riemannian geodesics of $(UM, D)$. Legendre singularities of sub-Riemannian geodesics are classified completely also for another Legendre projection from $UM$ to the space of Riemannian geodesics on $M$. The duality on Legendre singularities is observed related to the pendulum motion.