Lagrangian singularity

Summary

Take a Lagrangian submanifold of a cotangent bundle and project it to the base manifold. A Lagrangian singularity is a point where this projection fails to be a local diffeomorphism. The projection of a Lagrangian submanifold is called a Lagrangian map, and its singular points are the Lagrangian singularities. Their images form the caustic, which in optics is where rays focus. Arnold's theory classifies the generic Lagrangian singularities in low dimensions by the simple singularity types $A_k$, $D_k$ and $E_k$.