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Midpoint rule for variational integrators on Lie groups
Author(s) -
Saccon Alessandro
Publication year - 2009
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.2541
Subject(s) - variational integrator , midpoint , mathematics , lie group , integrator , symplectic geometry , manifold (fluid mechanics) , order (exchange) , function (biology) , mathematical analysis , geometry , physics , quantum mechanics , mechanical engineering , finance , voltage , economics , engineering , evolutionary biology , biology
The 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 ℝ n . 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 κ G ( 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.