One Identity, Three Languages: the Geometric Conservation Law, the Discrete Piola Identity, and Null Lagrangians

Abstract

Three communities keep the same identity under three names. Nonlinear elasticity calls it the Piola identity: div cof grad(phi) = 0 for every sufficiently smooth map. The calculus of variations calls it the null-Lagrangian property: the determinant (and each of its minors) is a Lagrangian density whose Euler–Lagrange equation vanishes identically. Computational fluid dynamics calls it the geometric conservation law (GCL): on curvilinear and moving grids the metric terms must satisfy a discrete identity, or else even uniform flow is not preserved. This note provides the translation dictionary between the three languages: it states that the surface-conservation half of the GCL is the discrete Piola identity, and that the success of conservative (curl-form) metric evaluation is due to the commutation of linear difference operators — on the discrete level exactly the mechanism that yields the continuous identity from the symmetry of mixed partials. The bridge between the two mathematical languages is known (the Piola identity is the Euler–Lagrange equation of a null Lagrangian); the note's contribution is the three-way table, bringing in the numerical community's identity, and a single deterministic verification suite covering all three languages (13/13, with counter-checks).