| Abstract | First variation formula for Lagrangian densities is a central element of the calculus of variations, that relates the differential of the action functional with both the Euler and Cartan forms, that are geometric objects, that is, as tensors, they do not depend on the choice of a coordinate system for the dependent or independent variables, or seen from a different viewpoint, they are invariant with respect to local automorphisms of these variables. First variation formula leads straightforward to Euler equations that characterize critical points of the action functional, as well as to Noether currents and conservation laws associated to any infinitesimal symmetry of the Lagrangian. This paper sets the scene for discrete variational problems on an abstract cellular complex that serves as discrete model of Rp and for the discrete theory of partial differential operators that are commom in the Calculus of Variations. A central result is the construction of a unique decomposition of certain partial difference operators into two components, one that is a vector bundle morphism and other one that leads to boundary terms. Application of this result to the differential of the discrete Lagrangian leads to unique discrete Euler and momentum forms not depending either on the choice of reference on the base lattice or on the choice of coordinates on the configuration manifold, and satisfying the corresponding discrete first variation formula. This formula leads to discrete Euler equations for critical points and to exact discrete conservation laws for infinitesimal symmetries of the Lagrangian density, with a clear physical interpretation. |
