Discrete Calculus of Variations for Quadratic Lagrangians

Abstract

The intent of this paper is to develop a framework for discrete calculus of variations with action densities involving a new class of discretization operators. We introduce first the generalized scale derivatives, study their regularity and state some Leibniz formulas. Then, we deduce the discrete EulerLagrange equations for critical points of sampled actions that we compare to existing versions. Next, we investigate the case of general quadratic lagrangians and provide two examples of such lagrangians. At last, we find nontrivial properties for the discretization of a quadratic nulllagrangian.