Discrete calculus of variations for quadratic lagrangians

TL;DR

This paper introduces a new framework for discrete calculus of variations with arbitrary discretization operators, deriving discrete Euler-Lagrange equations and analyzing their convergence to classical equations for quadratic Lagrangians.

Contribution

It develops a novel discrete calculus of variations framework and characterizes discretization operators ensuring convergence to classical Euler-Lagrange equations for quadratic Lagrangians.

Findings

Derived discrete Euler-Lagrange equations for sampled actions.

Characterized discretization operators that ensure convergence to classical equations.

Provided conditions under which discrete solutions approximate continuous ones.

Abstract

We develop in this paper a new framework for discrete calculus of variations when the actions have densities involving an arbitrary discretization operator. We deduce the discrete Euler-Lagrange equations for piecewise continuous critical points of sampled actions. Then we characterize the discretization operators such that, for all quadratic lagrangian, the discrete Euler-Lagrange equations converge to the classical ones.