Going through Villani's book on Optimal Transport, I came across the following setup: a Riemannian manifold $M$ and a Lagrangian $L : TM \times [0, 1] \to \mathbb{R}$ (So $L$ takes $(x, v, t)$ values as inputs). I'm wondering if I have the right notions of derivatives here. If $D_x L$ represents the spatial gradient, then we should have $$ D_x L(x_0, v_0, t_0)(w) = \dot\gamma(0), $$ where $\tilde{\gamma}(0) = x_0, \dot{\tilde{\gamma}}(0) = w$, and $\gamma(t) = L(\gamma(t), v_0, t_0)$. Is this correct?
I'm a little more confused about the notion of $D_vL$, where this represents the velocity gradient. I have just defined as follows: $$ D_v L(x_0, v_0, t_0)(w) = \dot\gamma(0), $$ where $\gamma(t) = L(x_0, v_0 + tw, t_0)$, because this seems to make sense to me.
I was wondering if anyone could 1) let me know if these are correct, and (perhaps more importantly) 2) whether or not there is a better way of thinking about these objects.
Much appreciation for any help.