For these increasing warps, the <square-root velocity function> is $Q(h)(t)=\sqrt{h'(t)}$. The <chain rule> gives
$$
Q(h_i\circ\gamma)(t)
=Q(h_i)(\gamma(t))\sqrt{\gamma'(t)}.
$$
Therefore
$$
\begin{aligned}
\lVert Q(h_i\circ\gamma)-Q(h_j\circ\gamma)\rVert_2^2
&=\int_0^1
|Q(h_i)(\gamma(t))-Q(h_j)(\gamma(t))|^2\gamma'(t)\,dt\\
&=\int_0^1|Q(h_i)(u)-Q(h_j)(u)|^2\,du,
\end{aligned}
$$
where the last equality again uses the <change of variables formula>. Taking square roots proves invariance under common right composition by $\gamma$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 225 / 3 / a / ii / Solution

{title2=$Q(f)$}

For an absolutely continuous real function $f$, its square-root velocity function is $Q(f)=f'/\sqrt{|f'|}$ where $f'\ne0$, with value zero where $f'=0$. Under an increasing reparameterization $\gamma$, it transforms as $Q(f\circ\gamma)=(Q(f)\circ\gamma)\sqrt{\gamma'}$.


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 Incoming links: Square-root velocity function