 Solution

Let $Q=(\sigma {\sigma }^T{)}^{-1}$ and use its induced inner product. The autonomous Lagrangian isIts conserved Hamiltonian isOn an infinite-duration fluctuation path $H=0$, so $\Vert \dot{\varphi }\Vert =\Vert a\Vert$. Expanding the action then givesSince metric arclength satisfies $ds=\Vert d\varphi \Vert =\Vert \dot{\varphi }\Vert dt=\Vert a\Vert dt$,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.