Lagrangian systems of conservation laws

We study the mathematical structure of 1D systems of conservation laws written in the Lagrange variable. Modifying the symmetrization proof of systems of conservation laws with three hypothesis, we prove that these models have a canonical formalism. These hypothesis are i) the entropy flux is zero, ii) Galilean invariance, iii) reversibility for smooth solutions. Then we study a family of numerical schemes for the solution of these systems. We prove that they are entropy consistent. We also prove from general considerations the symmetry of the spectrum of the Jacobian matrix.