On a class of mean fi eld solutions of the Monge problem for perfect and self-interacting systems
The Monge problem , , as reformulated by Kantorovich ,  is that of the transportation, at a minimum "cost", of a given mass distribu- tion from an initial to a final position during a given time interval. It is an optimal transport problem [28, sects. 1, 2]. Following the fluid mechanical solution provided by Benamou and Brenier for quadratic cost functions  ,[28, sects. 5.4, 8.1] and, by analogy with the fi xed end problem in Analytical Mechanics, Lagrangian formulations are needed to solve this boundary value problem in time. They are also needed to determine the Actions as time in- tegral of Lagrangians, that are measures of the "cost"of the transportations [4, proposition 1.1]. Four versions of explicit constructions of Lagrangians are proposed in section 3. They are associated to the Hamiltonians of perfect and self-interacting systems presented in section 2. These Hamiltonians are ex- pressed in function of pairs of the well known canonically conjugated Clebsch variables, namely mass densities and velocity potentials , . The fi rst version consists in the elimination of the velocity potentials as a function of the densities and their time derivatives by inversion of the continuity equations de- rived from given Hamiltonians. The second version consists in the elimination of the gradient of the velocity potentials from the continuity equations thanks to the introduction of vector valued applications such that their divergences give the mass densities. It turns out that, up to a sign factor, these vector elds are canonically conjugated to Euler velocity elds. The third version is a generalization in nD of Gelfand mass coordinate, a constant of the motion in 1 D , by the introduction of n-dimensional vector valued applications that enable to determine the mass densities as the determinant of their Jaco- bian matrices. Comparison of this set of mass coordinates with other sets of constants of the motion familiar in Fluid Dynamics is made in sub-section 3.3. Note that version two and three are identical for one-dimensional problems. The fourth version is based on the introduction of the Lagrangian coordinates that describe the characteristics of the different models and are parametrized by the former auxiliary vector fields. As illustrations, weak solutions of several models of Coulombian and Newtonian systems known in Plasma Physics and in Cosmology, respectively, with spherically symmetric boundary densities are given in section 4. However, and up to one exception given in the sub-section 3.3, calculations of the actions associated to these illustrations are not reported in this paper, nor the important analysis of the convexity-concavity properties of our Lagrangians. Lastly, and for the same models as those evoked above, a survey of past work concerning weak solutions of the Cauchy problem obeying the Hopf-Lax variational principle extended to negative time and having cor- related initial conditions is given in the Introduction as well as the derivation of the continuum fluid limit from many particle Hamiltonians.