---
tag: QKPMR9MC9W
kind: concept
name: Lagrangian singularity
msc: [53D12, 58K05]
sources:
  - url: https://en.wikipedia.org/wiki/Caustic_(mathematics)
---

## Summary

Take a [[Lagrangian submanifold|H8ACPTMX08]] 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|GMRYG4S3G0]], 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 $E_k$.

## Explanations

### L1 human

Think of a sheet lying in phase space, where each point records a position and a momentum.
Look at the sheet from above, so that you only see positions.
Where the sheet folds over, several of its points sit above the same position, and at the crease the view from above is singular.
Such a crease point is a Lagrangian singularity.
For example, the parabola $q = p^2$ in the $(q, p)$-plane, seen on the $q$-axis, folds at $p = 0$: positions $q > 0$ are hit twice and positions $q < 0$ not at all.