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The Integration of Angular Velocity

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Abstract

A common problem in physics and engineering is determination of the orientation of an object given its angular velocity. When the direction of the angular velocity changes in time, this is a nontrivial problem involving coupled differential equations. Several possible approaches are examined, along with various improvements over previous efforts. These are then evaluated numerically by comparison to a complicated but analytically known rotation that is motivated by the important astrophysical problem of precessing black-hole binaries. It is shown that a straightforward solution directly using quaternions is most efficient and accurate, and that the norm of the quaternion is irrelevant. Integration of the generator of the rotation can also be made roughly as efficient as integration of the rotation. Both methods will typically be twice as efficient as naive vector- or matrix-based methods. Implementation by means of standard general-purpose numerical integrators is stable and efficient, so that such problems can be readily solved as part of a larger system of differential equations. Possible generalization to integration in other Lie groups is also discussed.

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Acknowledgements

It is my pleasure to thank Scott Field, Larry Kidder, and Saul Teukolsky for useful conversations, as well as Nils Deppe for particularly enlightening discussions of stiffness and numerical integrators. I also appreciate Eva Zupan and Miran Saje for useful comments on their papers, and for pointing out more recent references. This project was supported in part by the Sherman Fairchild Foundation, and by NSF Grant Nos. PHY-1306125 and AST-1333129.

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  1. Cornell Center for Astrophysics and Planetary Science, Cornell University, Ithaca, NY, 14853, USA

    Michael Boyle

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  1. Michael Boyle
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Correspondence to Michael Boyle.

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Communicated by Leo Dorst.

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Boyle, M. The Integration of Angular Velocity. Adv. Appl. Clifford Algebras 27, 2345–2374 (2017). https://doi.org/10.1007/s00006-017-0793-z

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