Variation of a path

Definition

Let $M$ be a differential manifold and $\omega$ a path in $M$. A variation of $\omega$ is a map $\alpha :(-ϵ,ϵ)\times [0,1]\to M$ such that $\alpha (0,t)=\omega (t)\mathrm{\forall }t$.

For a given

Properties

Smooth variation

Further information: Smooth variation

A smooth variation is a variation for which the map $\alpha$ is smooth from the product manifold to $M$. Note that only a smooth path can admit a smooth variation,

Piecewise smooth variation

Further information: Piecewise smooth variation

A piecewise smooth variation is a variation for which we can find $o=s_1<s_2<\dots <s_r$ such that $\alpha$ restricted to $(-ϵ,ϵ)\times [s_{i-1},s_i]$ is smooth for $1\le i\le r$. Note that only a piecewise smooth path can admit a piecewise smooth variation.