Variational integrators for almost-integrable systems

Abstract

We construct several variational integrators—integrators based on a discrete variational principle—for systems with Lagrangians of the form L = LA + eLB, with \({\varepsilon \ll 1}\), where LA describes an integrable system. These integrators exploit that \({\varepsilon \ll 1}\) 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 \({\fancyscript{O} {(\varepsilon)}}\). 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.