Midpoint rule for variational integrators on Lie groups

Abstract

AbstractThe midpoint rule provides a standard method to obtain symmetric, symplectic, and second‐order accurate variational integrators for mechanical systems whose configuration manifold is the vector space ℝn. In this work, we discuss how to extend this rule to a generic finite‐dimensional Lie group G while retaining the same properties. We show that the function κG(g)=exp(½log(g)), g∈G plays a special role in the theory and, for G=SO(3), we give a compact formula to compute it. We also discuss sufficient conditions for the method to conserve momentum maps associated with left (or right) group actions.As an example, the variational integrator obtained from the midpoint rule is applied to simulating rigid body dynamics. The resulting integrator is compared with state‐of‐the‐art symmetric and second‐order accurate integrators for rigid body motion. Copyright © 2009 John Wiley & Sons, Ltd.