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  "body": "\n## Definition\n\nLet $Q$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$.\nLet $T^*Q$ 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.\nLet $L \\subset T^*Q$ be a [[Lagrangian submanifold|H8ACPTMX08]].\nThe *Lagrangian map* of $L$ is the restriction $\\pi|_L \\colon L \\to Q$, a smooth map between manifolds of the same dimension $n$.\nA 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 $n$.\n\n## Remarks\n\nExample: $Q = \\R$, $T^*Q = \\R^2$ with coordinates $(q, p)$, and $L = \\set{(p^2, p) : p \\in \\R}$.\nEvery curve in a symplectic surface is Lagrangian, so $L$ is Lagrangian.\nIn the parameter $p$ the Lagrangian map is $p \\mapsto p^2$, whose derivative $2p$ vanishes at $p = 0$.\nSo $(0, 0)$ is a Lagrangian singularity, of fold type $A_2$.\n\nNon-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 $Q$.\nThe 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.\n\nLagrangian singularities are studied up to Lagrangian equivalence: symplectomorphisms of $T^*Q$ that map fibres of $\\pi$ to fibres.\nLocally every Lagrangian submanifold is given by a generating family $F(q, y)$ with $y \\in \\R^m$, as $L = \\set{(q, \\partial_q F) : \\partial_y F = 0}$.\nIts 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.\n",
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