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 = 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.

14 pages, 4 figures. Version 2: added informative example; as accepted by Celestial Mechanics and Dynamical Astronomy

References in corpus (3)

• High order symplectic integrators for perturbed Hamiltonian systems
• Pseudo-High-Order Symplectic Integrators
• Variational Integrators for the Gravitational N-Body Problem

Cited by in corpus (4)

• Slimplectic Integrators: Variational Integrators for General Nonconservative Systems
• Practical Perspectives on Symplectic Accelerated Optimization
• Variational integrators for perturbed non-canonical Hamiltonian systems
• Analytical theory for highly elliptical orbits including time-dependent perturbations