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  "body": "\n## Definition\n\nLet $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] and let $a \u003c b$.\nFor a continuous, piecewise smooth path $\\omega \\colon [a, b] \\to M$, the *energy* of $\\omega$ is\n$$E(\\omega) = \\frac{1}{2} \\int_a^b g\\big(\\dot\\omega(t), \\dot\\omega(t)\\big) \\, dt.$$\nFor $p, q \\in M$, the *energy functional on paths* from $p$ to $q$ is the function $E$ on the set of such paths with $\\omega(a) = p$ and $\\omega(b) = q$.\n\n## Remarks\n\nSome authors omit the factor $\\frac{1}{2}$.\nWith the [[length|0TH78S7GBZ]] $L(\\omega) = \\int_a^b \\abs{\\dot\\omega} \\, dt$, the Cauchy–Schwarz inequality gives $L(\\omega)^2 \\le 2 (b - a) E(\\omega)$, with equality if and only if $\\omega$ has constant speed.\nThe critical points of $E$ among paths with fixed ends are the geodesics from $p$ to $q$ parametrized on $[a, b]$.\nA minimizer of $E$ is a length-minimizing geodesic with constant speed.\nTo apply Morse theory one extends $E$ to paths of Sobolev class $H^1$, which form a Hilbert manifold.\n\nExample: in $\\R^n$ with the Euclidean metric, the path $\\omega(t) = p + t(q - p)$ on $[0, 1]$ has $E(\\omega) = \\frac{1}{2} \\abs{q - p}^2$, and $L(\\omega)^2 = 2 E(\\omega)$, so it has constant speed.\n\nNon-example: the reparametrization $\\sigma(t) = p + t^2 (q - p)$ of the same segment has $E(\\sigma) = \\frac{2}{3} \\abs{q - p}^2 \u003e \\frac{1}{2} \\abs{q - p}^2$.\nIt has the same length but is not a critical point of $E$, since its speed is not constant.\n",
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