Variational Integrators for Almost-Integrable Systems

TL;DR

This paper develops variational integrators tailored for almost-integrable systems, leveraging their structure to enhance accuracy and simplify error analysis, including a novel method that eliminates errors at order epsilon.

Contribution

The paper introduces new variational integrators for systems with small perturbations, simplifying their construction and analysis, and presents a novel implicit method that removes leading-order errors.

Findings

Integrators are equivalent to known symplectic methods but easier to analyze.

A new weighted averaging method removes all O(epsilon) errors.

The implicit method can outperform traditional simulation techniques.

Abstract

We construct several variational integrators--integrators based on a discrete variational principle--for systems with Lagrangians of the form L = L_A + epsilon L_B, with epsilon << 1, where L_A describes an integrable system. These integrators exploit that epsilon << 1 to increase their accuracy by constructing discrete Lagrangians based on the assumption that the integrator trajectory is close to that of the integrable system. Several of the integrators we present are equivalent to well-known symplectic integrators for the equivalent perturbed Hamiltonian systems, but their construction and error analysis is significantly simpler in the variational framework. One novel method we present, involving a weighted time-averaging of the perturbing terms, removes all errors from the integration at O(epsilon). This last method is implicit, and involves evaluating a potentially expensive time-integral, but for some systems and some error tolerances it can significantly outperform traditional simulation methods.