On a Class of Mean Field Solutions of the Monge Problem for Perfect and Self-Interacting Systems

The Monge problem (Monge 1781; Taton 1951), as reformulated by Kantorovich (2006a, 2006b) is that of the transportation at a minimum “cost” of a given mass distribution from an initial to a final position during a given time interval. It is an optimal transport problem (Villani, 2003, sects. 1, 2). Following the fluid mechanical solution provided by Benamou and Brenier (2000) for quadratic cost functions (Villani, 2003, sects. 5.4, 8.1), Lagrangian formulations are needed to solve this boundary value problem in time and to determine the Actions as time integral of Lagrangians that are measures of the “cost” of the transportations (Benamou and Brenier, 2000, prop. 1.1). Given canonical Hamilltonians of perfect and self-interacting systems expressed in function of mass densities and velocity potentials, four versions of explicit constructions of Lagrangians, with their corresponding generalized coordinates, are proposed: elimination of the velocity potentials as a function of the densities and their time derivatives by inversion of the continuity equations; elimination of the gradient of the velocity potentials from the continuity equations thanks to the introduction of vector fields such that their divergences give the mass densities; generalization in nD of Gelfand mass coordinate (1963) by the introduction of n-dimensional vector fields such that the determinant of their Jacobian matrices give the mass densities; and, last, introduction of the Lagrangian coordinates that describe the characteristics of the different models and are parametrized by the former auxiliary vector fields. Using this version, weak solutions of several models of Coulombian and Newtonian systems known in Plasma Physics and in Cosmology, with spherically symmetric boundary densities, are given as illustrations.