edit: from Sangchul Lee's comments, it appears my definition was initially inaccurate so it has been corrected. In physics one finds the following definition of the action density:
$$ \mathcal{S}[\Delta t,\Delta x,\Delta y,\Delta z]=\int_{\Delta z}\int_{\Delta y}\int_{\Delta z}\int_{\Delta t} \mathcal{L}[t,x,y,z]dtdxdydz $$
Here, as a challenge, I am interested in defining the same concept but in the reverse direction:
$$ \mathcal{L}[t,x,y,z] = D[\mathcal{S}[\Delta t,\Delta x,\Delta y,\Delta z]] $$
where $D$ is a suitable derivative, and where $\Delta t,\Delta x,\Delta y,\Delta z$ are pairs of real numbers.
What is $D$ in this case? If I take the total derivative I do not get the 4-form differential such $dtdxdydz$. If I instead take the gradient, then $\mathcal{L}$ is a vector, however, it must be a scalar. Finally, the last candidate I know is the directional derivative which produces a scalar --- So, is the Lagrangian the directional derivative of the action (the dependence on a path seems awfully specific)?