---
tag: 43R0MPN4T1
kind: definition
name: Lagrangian singularity
msc: [53D12, 58K05]
defines: QKPMR9MC9W
uses: [F45VQ47GPV, CAN490EPRT, H8ACPTMX08, 6P9PXQDSEN]
sources:
  - url: https://en.wikipedia.org/wiki/Caustic_(mathematics)
---

## Definition

Let $Q$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$.
Let $T^*Q$ be its cotangent bundle with the canonical [[symplectic form|CAN490EPRT]] $\omega = \sum_{i=1}^n dp_i \wedge dq_i$ in local coordinates $(q, p)$, and let $\pi \colon T^*Q \to Q$ be the bundle projection.
Let $L \subset T^*Q$ be a [[Lagrangian submanifold|H8ACPTMX08]].
The *Lagrangian map* of $L$ is the restriction $\pi|_L \colon L \to Q$, a smooth map between manifolds of the same dimension $n$.
A point $x \in L$ is a *Lagrangian singularity* if the [[differential|6P9PXQDSEN]] $d(\pi|_L)_x \colon T_x L \to T_{\pi(x)} Q$ is not invertible, that is, if its rank is less than $n$.

## Remarks

Example: $Q = \R$, $T^*Q = \R^2$ with coordinates $(q, p)$, and $L = \set{(p^2, p) : p \in \R}$.
Every curve in a symplectic surface is Lagrangian, so $L$ is Lagrangian.
In the parameter $p$ the Lagrangian map is $p \mapsto p^2$, whose derivative $2p$ vanishes at $p = 0$.
So $(0, 0)$ is a Lagrangian singularity, of fold type $A_2$.

Non-example: the zero section $L_0 = \set{p = 0}$, or more generally the graph $\set{(q, dS(q))}$ of the differential of a smooth function $S$ on $Q$.
The projection restricted to such a graph is a diffeomorphism onto $Q$, so its differential is invertible at every point and there is no Lagrangian singularity.

Lagrangian singularities are studied up to Lagrangian equivalence: symplectomorphisms of $T^*Q$ that map fibres of $\pi$ to fibres.
Locally every Lagrangian submanifold is given by a generating family $F(q, y)$ with $y \in \R^m$, as $L = \set{(q, \partial_q F) : \partial_y F = 0}$.
Its Lagrangian singularities then lie over the points $q$ where $F(q, \cdot)$ has a degenerate critical point, which links the theory to the [[bifurcation set|JV45S0KQE6]] of catastrophe theory.