= Solution
Let $Q=(\sigma\sigma^T)^{-1}$ and use its induced inner product. The autonomous Lagrangian is
$$
L=\frac12\|\dot\phi-a\|^2.
$$
Its conserved Hamiltonian is
$$
H=\frac12(\|\dot\phi\|^2-\|a\|^2).
$$
On an infinite-duration fluctuation path $H=0$, so $\|\dot\phi\|=\|a\|$. Expanding the action then gives
$$
I=\int\left(\|\dot\phi\|^2-a\cdot\dot\phi\right)dt.
$$
Since metric arclength satisfies $ds=\|d\phi\|=\|\dot\phi\|dt=\|a\|dt$,
$$
\boxed{I[\phi]=\int_\phi\|a\|\,ds-a\cdot d\phi}.
$$
This is the <geometric minimum action> representation and is independent of the speed used to parametrize the path.
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