Lagrangian singularity

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 of catastrophe theory.