  Given a compact K\"ahler manifold, the space $\mathcal H$ of its (relative)
K\"ahler potentials is an infinite dimensional Fr\'echet manifold, on which
Mabuchi and Semmes have introduced a natural connection $\nabla$. We study
certain Lagrangians on $T\mathcal H$, in particular Finsler metrics, that are
parallel with respect to the connection. We show that geodesics of $\nabla$ are
paths of least action; under suitable conditions the converse also holds; and
prove a certain convexity property of the least action. This generalizes
earlier results of Calabi, Chen, and Darvas.
