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.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 353 / 4 / c / i / Solution

On a zero-Hamiltonian infinite-time fluctuation path, the Freidlin--Wentzell action can be parametrized independently of time as $I=\int\|a\|ds-a\mathbin\cdot d\phi$ in the diffusion metric.


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 Incoming links: Geometric minimum action