= Geometric minimum action
<Geometric minimum action> removes the time parametrization when minimizing the <Freidlin--Wentzell action> over both a path and its travel time. With a positive-definite diffusion <covariance> $D(\phi)$, write $\langle u,v\rangle_D=u^TD^{-1}v$ and $\|u\|_D=\sqrt{\langle u,u\rangle_D}$. For an absolutely continuous curve $\phi(s)$, the free-time geometric cost is
$$
\widehat I(\phi)=\int\left[\|a(\phi)\|_D\|\phi'(s)\|_D-\langle a(\phi),\phi'(s)\rangle_D\right]ds.
$$
It is invariant under increasing changes of the curve parameter. This is a free-time cost used in the <quasipotential>, not a replacement for the action at a prescribed finite arrival time.<The geometric minimum action method: A least action principle on the space of curves by Matthias Heymann and Eric Vanden-Eijnden>[Sections 1.2–1.3, free-time variational problem and geometric reformulation]{ref}
To derive the integrand, set $r=dt/ds>0$ in the time action. Its density becomes
$$
\frac12\left(\frac{\|\phi'\|_D^2}{r}+r\|a\|_D^2-2\langle a,\phi'\rangle_D\right).
$$
Minimizing over $r$ uses $A/r+rB\geq2\sqrt{AB}$ and gives the geometric expression. Where both <norms> are positive, the optimal speed satisfies $r=\|\phi'\|_D/\|a\|_D$. At equilibria the infimum may require an infinite travel time, explaining the usefulness of an arc-length-like parameter.
For gradient drift $a=-\nabla U$ and $D=I$, an uphill gradient curve has cost $2[U(\mathrm{end})-U(\mathrm{start})]$, while a downhill deterministic trajectory has zero cost. The factor two corresponds to noise $\sqrt\varepsilon\,dW$; changing to $\sqrt{2\varepsilon}\,dW$ changes the action normalization.
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