First variation formula for energy

Shrink to a closed parameter interval $[-\delta ,\delta ]\subset (-\epsilon ,\epsilon )$. On each compact strip, smoothness and [F4] allow differentiation under the integral. Metric compatibility then gives ${\frac{d}{ds}\mid }_0\frac{1}{2}{\int }_{t_{j-1}}^{t_j}g({\mathrm{\partial }}_t\alpha ,{\mathrm{\partial }}_t\alpha )dt={\int }_{t_{j-1}}^{t_j}g(D_s{\mathrm{\partial }}_t\alpha ,T)dt\mathrm{.}$

In a coordinate chart along any smooth part of a strip, writing $x^k=x^k\circ \alpha$ gives $(D_s{\mathrm{\partial }}_t\alpha {)}^k={\mathrm{\partial }}_s{\mathrm{\partial }}_t{x}^k+{\mathrm{\Gamma }}^k{}_{i\mathrm{\ell }}{\mathrm{\partial }}_s{x}^i{\mathrm{\partial }}_t{x}^{\mathrm{\ell }},$ while $(D_t{\mathrm{\partial }}_s\alpha {)}^k={\mathrm{\partial }}_t{\mathrm{\partial }}_s{x}^k+{\mathrm{\Gamma }}^k{}_{\mathrm{\ell }i}{\mathrm{\partial }}_t{x}^{\mathrm{\ell }}{\mathrm{\partial }}_s{x}^i\mathrm{.}$ Equality of mixed partials and the symmetry in [F3] prove $D_s{\mathrm{\partial }}_t\alpha =D_t{\mathrm{\partial }}_s\alpha$; the coordinate identities agree on overlaps, so this holds on every strip.

At $s=0$, steps 1.1--1.2 and the metric product rule give $g(D_{t}V,T)=\frac{d}{dt}g(V,T)-g(V,D_{t}T)\mathrm{.}$ Applying [F4] and summing over the finite common subdivision therefore yields ${\frac{d}{ds}\mid }_{0}E({\gamma }_s)={\sum }_{j=1}^m(g(V(t_j),T(t_j^{-}))-g(V(t_{j-1}),T(t_{j-1}^{+}))-{\int }_{t_{j-1}}^{t_j}g(V,D_{t}T)dt)\mathrm{.}$

The outer boundary contributions in step 2.1 are $g(V(b),T(b^{-}))-g(V(a),T(a^{+}))$. Continuity of $V$ makes the two contributions at an interior $t_j$ equal to $g(V(t_j),T(t_j^{-})-T(t_j^{+}))=-g(V(t_j),T(t_j^{+})-T(t_j^{-}))$. This is the asserted formula. When $m=1$ no corner occurs, giving the smooth formula.

If the variation fixes the endpoints then $V(a)=V(b)=0$, but moving endpoints retain both displayed terms. A constant central curve makes $T=D_{t}T=0$, so every term vanishes. In dimension zero all terms vanish; dimension one uses the same calculation. An empty $M$ admits no such curve. The hypothesis $a<b$ and common finite subdivision exclude an empty interval and infinite summation; one-sided endpoint and corner derivatives are precisely those in [F3]. The Levi--Civita connection is uniquely constructed from the supplied metric by [F2], and every remaining operation is finite or pointwise, so no choice axiom is used.