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  "body": "\n## Definition\n\nLet $a \u003c b$ and $A, B \\in \\R$, and let $J(y) = \\int_a^b L(x, y, y') \\, dx$ be an [[integral functional|JRQZ803Q5Y]] with a continuous Lagrangian $L \\colon [a, b] \\times \\R \\times \\R \\to \\R$.\nIn the fixed-endpoint problem for $J$, a function $y \\colon [a, b] \\to \\R$ is *admissible* if:\n\n1. $y \\in C^1([a, b])$, that is, $y$ is [[continuously differentiable|0VW90BBSJ4]] on $[a, b]$;\n2. $y(a) = A$;\n3. $y(b) = B$.\n\nThe set $\\mathcal{A}$ of admissible functions is the *admissible class*.\nMore generally, the admissible class of a variational problem is the set of functions in the domain of the functional that satisfy all its constraints.\nIts elements are the admissible functions.\n\n## Remarks\n\nExample: for $a = 0$, $b = 1$, $A = 0$, $B = 1$, the function $y(x) = x^2$ is admissible: it is $C^1$, $y(0) = 0$ and $y(1) = 1$.\nNon-example: for $a = 0$, $b = 1$, $A = B = 1$, the function $y(x) = \\abs{x - 1/2} + 1/2$ satisfies $y(0) = y(1) = 1$ but is not differentiable at $1/2$, so it fails clause 1.\nMany texts use the larger class of piecewise $C^1$ functions, which admits such broken curves.\nIf $y$ is admissible and $\\eta \\in C^1([a, b])$ has $\\eta(a) = \\eta(b) = 0$, then $y + t\\eta$ is admissible for every $t \\in \\R$; such $\\eta$ are the admissible variations.\n",
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