Energy functional on paths

Remarks

Some authors omit the factor $\frac{1}{2}$. With the length $L(\omega) = \int_a^b \abs{\dot\omega} \, dt$, the Cauchy–Schwarz inequality gives $L(\omega)^2 \le 2 (b - a) E(\omega)$, with equality if and only if $\omega$ has constant speed. The critical points of $E$ among paths with fixed ends are the geodesics from $p$ to $q$ parametrized on $[a, b]$. A minimizer of $E$ is a length-minimizing geodesic with constant speed. To apply Morse theory one extends $E$ to paths of Sobolev class $H^1$, which form a Hilbert manifold.

Example: 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.

Non-example: the reparametrization $\sigma(t) = p + t^2 (q - p)$ of the same segment has $E(\sigma) = \frac{2}{3} \abs{q - p}^2 > \frac{1}{2} \abs{q - p}^2$. It has the same length but is not a critical point of $E$, since its speed is not constant.