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  "body": "\n## Definition\n\nLet $Q$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$.\nLet $\\pi \\colon T^*Q \\to Q$ be its cotangent bundle with the canonical symplectic form, and let $L \\subset T^*Q$ be a [[Lagrangian submanifold|H8ACPTMX08]].\nLet $\\Sigma_L \\subset L$ be the set of [[Lagrangian singularities|QKPMR9MC9W]] of $L$, the points where the differential of $\\pi|_L \\colon L \\to Q$ has rank less than $n$.\nThe *caustic* of $L$ is the set\n$$C_L = \\pi(\\Sigma_L) \\subset Q$$\nof critical values of the Lagrangian map $\\pi|_L$.\n\n## Remarks\n\nExample: $Q = \\R$ and $L = \\set{(p^2, p) : p \\in \\R} \\subset T^*\\R = \\R^2$.\nThe map $\\pi|_L$ is $p \\mapsto p^2$, singular only at $p = 0$, so $\\Sigma_L = \\set{(0, 0)}$ and $C_L = \\set{0}$.\nOver $q \u003e 0$ lie two points of $L$, and over $q \u003c 0$ none, so the number of preimages jumps at the caustic.\n\nNon-example: if $L$ is the graph of $dS$ for a smooth function $S$ on $Q$, then $\\pi|_L$ is a diffeomorphism, $\\Sigma_L$ is empty, and the caustic is empty.\n\nIn geometric optics, $Q$ is space and $L$ is the set of positions and directions of the rays of a beam.\nThen $C_L$ is the envelope of the family of rays, where the rays focus.\nIf $L$ is given locally by a generating family $F(q, y)$, the caustic is the [[bifurcation set|JV45S0KQE6]] of the family $y \\mapsto F(q, y)$ with $q$ as control.\n",
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