= Fixed-endpoint variation of a curve
= Fixed-endpoint variations of a curve
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= Proper variation of a curve
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= Proper variations of a curve
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A fixed-endpoint variation of a <curve> $\gamma:[a,b]\to S$ is a smooth family $\Gamma(s,t)$ with $\Gamma(0,t)=\gamma(t)$, $\Gamma(s,a)=\gamma(a)$ and $\Gamma(s,b)=\gamma(b)$. Its <variation vector field> $V(t)=\partial_s\Gamma(0,t)$ vanishes at the endpoints. Consequently integration by parts in the first variation of <energy of a curve> has no boundary contribution, and stationarity is characterized by the <geodesic equation>.
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