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 + εL B , with ε ≪ 1, where L A describes an integrable system. These integrators exploit that ε ≪ 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 (ε). 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.

“Slimplectic” Integrators: Variational Integrators for General Nonconservative Systems

“…Similar results can be obtained for Wisdom-Holman-type mappings by splitting the nonconservative action into integrable and perturbative terms (see e.g Farr 2009…”

Properties of Hamiltonian variational integrators

“…A variational integrator for such a system was proposed in [2] using a discrete Lagrangian formulation, which drew inspiration from the kick-drift-kick leapfrog method (see [18]). We will discuss the Lagrangian formulation (hereafter referred to as the averaged Lagrangian) and in addition construct an analogous method in terms of a discrete right Hamiltonian (referred to as the averaged Hamiltonian).…”

“…To clarify notation we will be assuming that (q 0 , p 0 ) are the initial conditions for both implementations, and we introduce (q 1,L d , p 1,L d ) and (q 1,H + d , p 1,H + d ) to denote the respective numerical approximations after one timestep. The method proposed in [2] used a discrete Lagrangian of the form,…”

“…) to denote the respective numerical approximations after one timestep. The method proposed in [2] used a discrete Lagrangian of the form,…”

“…This can be viewed as the solution at time T of the Type II Hamilton-Jacobi equation, ∂S 2 (q 0 , p, t) ∂t = H ∂S 2 ∂p , p , (2) which more generally describes the Type II generating function which generates the time-t Hamiltonian flow map,…”