Abstract
Systems of particles are naturally described in a Lagrangian way : the unknowns (positions, velocities...) are related to the entities, numbered once and for all, and the Euclidean frame in this Lagrangian framework is very well adapted to the laws of classical mechanics.
When describing the motion of a continuum of matter on a macroscopic scale, the Eulerian framework is the most natural. This is the approach used to write partial differential equations for continuous media, for which the functional spaces (typically Sobolev spaces) are Eulerian-inspired.
The framework of quadratic optimal transport, which induces a metric based on matter displacements (variations / horizontal velocities in space-time), makes it possible to inject a Lagrangian character into the macroscopic description, to recover the Hilbertian character of classical mechanics, and thus to use demonstration techniques close to those used for discrete particle systems.
We will illustrate these considerations through microscopic and macroscopic models of crowd movements, as well as models of granular media / pressureless gas with congestion constraint, and we will show that this approach to optimal transport is only partially Lagrangian, which gives the Wasserstein space particular properties and prevents a perfect transposition from the micro to the macro frame.
