is understood as follows: a tangent vector $h\in T_\sigma\mathcal P^\infty(M)$ is represented by a potential $\psi\in C^\infty(M;\mathbb R)$ through $h=-\operatorname{div}_g(\sigma\nabla\psi)$, with $\psi$ unique up to an additive function constant on each connected component of $M$, and the formal Levi-Civita acceleration of a velocity potential $\phi_t$ is represented by the gradient projection of $\partial_t\nabla\phi_t+\nabla^M_{\nabla\phi_t}\nabla\phi_t$, where $\nabla^M$ is the Levi-Civita connection of $(M,g)$. Then the curve
is a formal $W_2$-geodesic in Otto's Riemannian structure with velocity field $v_t=\nabla\phi_t$ if and only if, after replacing $\phi_t$ by an additive gauge which is constant on each connected component of $M$ and depends smoothly on $t$, the pair $(\rho_t,\phi_t)$ satisfies, for every $t\in(0,1)$,