Discrete calculus of variations for quadratic lagrangians
Provides a theoretical foundation for discrete variational problems, relevant to numerical analysts and physicists working on discretization of continuous systems.
This paper develops a discrete calculus of variations framework for actions with densities involving arbitrary discretization operators, deriving discrete Euler-Lagrange equations and characterizing operators that ensure convergence to classical equations for quadratic Lagrangians.
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.