Common variational principles for Euler, Einstein and Schroedinger equations

In spite of their very different natures, the Euler, Einstein and Schroedinger Equations share very similar variational structures in strong connection with Optimal Transport Theory, involving density fields that are respectively real, matrix and complex-valued. A key point of the analysis is to write all these equations as quadratic system ofconservation laws, in particular thanks to the Madelung transform in the case of Schroedinger's equation.