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  "body": "\n## Definition\n\nLet $L(x, y, p)$ be a $C^2$ function on $\\R \\times \\R \\times \\R$, with [[integral functional|JRQZ803Q5Y]] $J(y) = \\int L(x, y, y') \\, dx$.\nLet $D \\subseteq \\R^2$ be a simply connected open set.\nA *field of extremals* of $J$ on $D$ is a family of functions $y = \\varphi(x, c)$, with $c$ in an open interval $I$, such that:\n\n1. for each $c \\in I$, the function $x \\mapsto \\varphi(x, c)$ solves the [[Euler–Lagrange equation|PS0VV149ET]] of $J$;\n2. through each point $(x, y) \\in D$ passes exactly one curve $y = \\varphi(x, c)$ of the family;\n3. the parameter $c(x, y)$ of that curve is a [[continuously differentiable|0VW90BBSJ4]] function on $D$.\n\nThe *slope function* of the field is $p(x, y) = \\partial_x \\varphi(x, c(x, y))$.\nAn extremal $y_0$ is *embedded* in the field if $y_0 = \\varphi(\\cdot, c_0)$ for some $c_0 \\in I$.\n\n## Remarks\n\nExample: for $L = \\sqrt{1 + p^2}$ the extremals are straight lines.\nThe family $\\varphi(x, c) = x + c$, $c \\in \\R$, is a field on $D = \\R^2$, with $c(x, y) = y - x$ and slope function $p = 1$.\n\nNon-example: the family $\\varphi(x, c) = c x$, $c \\in \\R$, of lines through the origin is not a field on $\\R^2$.\nAll its curves pass through $(0, 0)$, and no curve passes through the points $(0, y)$ with $y \\ne 0$.\nOn the half-plane $x \u003e 0$ it is a field, with $c = y / x$; it is then a *central field*, whose curves all start at one point on the boundary.\n\nLet an extremal $y_0$ on $[a, b]$ satisfy $L_{pp} \u003e 0$ along it, with no point conjugate to $a$ in $(a, b]$.\nThen it can be embedded in a field on a neighbourhood of its graph.\n",
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