Error analysis of forced discrete mechanical systems

TL;DR

This paper provides an error analysis of variational integrators for forced mechanical systems, establishing that the integrator's contact order matches that of the discretization of the Lagrangian and force, with implications for system accuracy.

Contribution

It introduces a rigorous error analysis framework for forced variational integrators, linking discretization contact order to integrator accuracy for forced mechanical systems.

Findings

Integrators preserve contact order of the discretization.

Existence of flows for discrete forced systems is established.

Contact order of flows matches that of the discretization.

Abstract

The purpose of this paper is to perform an error analysis of the variational integrators of mechanical systems subject to external forcing. Essentially, we prove that when a discretization of contact order $r$ of the Lagrangian and force are used, the integrator has the same contact order. Our analysis is performed first for discrete forced mechanical systems defined over $TQ$, where we study the existence of flows, the construction and properties of discrete exact systems and the contact order of the flows (variational integrators) in terms of the contact order of the original systems. Then we use those results to derive the corresponding analysis for the analogous forced systems defined over $Q\times Q$.