Lagrangian field functions have units of energy density. Energy is usually defined in scalar terms as the dot product of two vectors. In this question two vectors are being compared. So I suspect the answer is no.

The other choice seems to be possible with gravity potential energy density being compared to kinetic energy density which is described by a scalar divergence. For agreement of units the related momentum vector needs a multiplier to be compatible with the divergence dot product method and produce an energy density.  Then the vector would have units of velocity times momentum per cross sectional area.

Divergence is not essential since energy can be derived by dot products.

dE = v · dp

 In which case the same volume element applies to both sides of the equation to make energy density for the Lagrangian field method.